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USCHEP/0304ib2 hep-th/0304005
UT-03-08 April, 2003
Fermionic Ghosts in Moyal String Field Theory
I. Barsa,∗, I. Kishimotob,† and Y. Matsuob,‡
a) Department of Physics and Astronomy,
University of Southern California, Los Angeles, CA 90089-0484, USA
b) Department of Physics, Faculty of Science, University of Tokyo
Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japan
Abstract
We complete the construction of the Moyal star formulation of bosonic open string field
theory (MSFT) by providing a detailed study of the fermionic ghost sector. In particular, as in
the case of the matter sector, (1) we construct a map from Witten’s star product to the Moyal
product, (2) we propose a regularization scheme which is consistent with the matter sector and
(3) as a check of the formalism, we derive the ghost Neumann coefficients algebraically directly
from the Moyal product. The latter satisfy the Gross-Jevicki nonlinear relations even in the
presence of the regulator, and when the regulator is removed they coincide numerically with
the expression derived from conformal field theory. After this basic construction, we derive a
regularized action of string field theory in the Siegel gauge and define the Feynman rules. We
give explicitly the analytic expression of the off-shell four point function for tachyons, including
the ghost contribution. Some of the results in this paper have already been used in our previous
publications. This paper provides the technical details of the computations which were omitted
there.
∗e-mail address: [email protected]†e-mail address: [email protected]‡e-mail address: [email protected]
Contents
1 Introduction 2
2 Moyal’s star from Witten’s star in fermionic ghost sector 6
2.1 Half string formalism and regularization . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.1 Dirichlet at end point, Neumann at midpoint (DN) . . . . . . . . . . . . . . 9
2.1.2 Dirichlet at end point, Dirichlet at midpoint (DD) . . . . . . . . . . . . . . . 10
2.1.3 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.4 GL(N |N) supergroup property of the regulator . . . . . . . . . . . . . . . . 14
2.2 Moyal ⋆ from Witten’s ∗ for fermionic modes . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Moyal ⋆ product in the bc ghost sector . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4.1 Oscillators on the fields in MSFT . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4.2 Oscillator as a field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4.3 L0 and L0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3 Monoid and Neumann coefficients 26
3.1 Monoid structure for gaussian elements . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Neumann coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3 Numerical comparison of Neumann coefficients . . . . . . . . . . . . . . . . . . . . . 34
4 Applications 35
4.1 Regularized MSFT action and equation of motion . . . . . . . . . . . . . . . . . . . . 35
4.2 Computing Feynman graphs including fermionic ghost sector . . . . . . . . . . . . . 37
5 Discussion 42
A Brief review of MSFT in matter sector 43
A.1 Half-string for cosine modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
A.1.1 Dirichlet at midpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
A.1.2 Neumann at midpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
1
A.1.3 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
A.2 Some results in matter sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
A.2.1 Oscillators in MSFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
A.2.2 Butterfly projector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
A.2.3 L0 and L0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
A.2.4 n-string vertex and Neumann coefficients . . . . . . . . . . . . . . . . . . . . 50
B Derivation of regularized matrix formula 53
C bc-ghost sector in position space 54
D Moyal ⋆ for (DD)b (NN)c split strings 56
E Ghost butterfly projector with even modes 57
F Algebra of gaussian operators 58
G Neumann coefficients 60
G.1 Neumann coefficients from CFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
G.2 Ratios of MSFT-regulated and CFT Neumann coefficients . . . . . . . . . . . . . . . 61
H Twist and SU(1, 1) in the Siegel gauge 62
1 Introduction
Witten’s open bosonic string field theory [1], its operator version [2], and its split string reformu-
lations [3, 4, 5, 6, 7], led to the development of the Moyal star formulation of string field theory
(MSFT) during the past two years [7]-[12]. This formulation has the following features:
1. The star product in Witten’s string field theory is mapped to the Moyal star product [7]
after an appropriate change of variables. The string field A (x, xe.pe) in the new basis is
a function of the string midpoint x and the phase space (xe, pe) of even string modes e =
2, 4, 6, · · · . While Witten’s star product, written in terms of Neumann coefficients [2], is
very complicated to manipulate in computations, the Moyal star product in MSFT, which is
diagonal in mode space labelled by e, is the simplest form of the star product that occurs
in standard noncommutative geometry. This feature vastly simplifies the structure of string
2
field theory and is very helpful in explicit computations.
2. The change of variables introduces a set of simple but infinite matrices Teo, Roe, vo, we, labelled
by e = 2, 4, 6, · · · , and o = 1, 3, 5, · · · , which contain basic information about even (e) and odd
(o) string modes. These matrices obey a matrix algebra that has an associativity anomaly,
which in turn feeds into an associativity anomaly among string fields [8]. The origin of the
anomaly is the infinite number of string modes that cause the appearance of ambiguous terms
of the form ∞/∞. In order to resolve this problem, we proposed [8, 9] a regularization by
truncating the number of the oscillators to finite 2N , and defined a deformed set of finite
matrices T,R, v, w as functions of the oscillator frequencies κe, κo of the 2N modes. After
such regulation, the associativity is restored and all manipulations become well-defined. The
original open string field theory is restored by taking the original frequencies, κe = e, κo = o,
and the large N limit at the end of the computation. Through explicit computation of
specific examples, it has been shown that this regulation procedure correctly reproduces
results computed independently in conformal string theory.
3. In the regularized basis, we computed the Neumann coefficients analytically by using only
the Moyal product. These coefficients are not needed for computations in MSFT, but they
provide a check of MSFT relative to the operator formulation given in [2]. We have shown
that the Neumann coefficients derived in the regularized MSFT framework satisfy the Gross-
Jevicki nonlinear relations, and thus provide a generalization of Neumann coefficients for any
set of frequencies κe, κo and any N. This provided the first consistency check of MSFT [9].
Furthermore, we found that the Neumann coefficients for any n-point vertex are all simple
functions of a single matrix teo = κ1/2e Teoκ
−1/2o . Diagonalizing the matrix t diagonalizes all
the Neumann coefficients simultaneously for all n-point vertices [9]. At large N our diagonal
form agrees with the one given in [13][14], and explains in particular why there is Neumann
spectroscopy for the 3-point vertex [13], and generalizes it to any n-point vertex [9].
4. One of the nice features of MSFT is that the star product is diagonal in mode space (i.e.
independent and same for each mode). The only cost for this simplification is that the
kinetic term given by the Virasoro operator L0 becomes off-diagonal in mode space (xe, pe)
[9]. However, this has not hindered explicit computations. In particular, we have derived
the Feynman rules, including the propagator (L0 − 1)−1 , and shown that we can evaluate
efficiently and explicitly the Feynman graphs in open string field theory [10].
5. The off-diagonal part of the kinetic term depends on a specific combination of momentum
modes, namely p = (1 + ww)−1/2∑ewepe which we refer to as the “anomalous midpoint
mode” [8][11]. This mode appears in the kinetic operator in the form L0 = γ + · · · , withγ ∼ p2, while the remaining part of the kinetic term is diagonal in mode space. We named
the term γ the “midpoint correction” . We found that if it were not for this midpoint
correction, the rest of the kinetic plus interaction terms would define a theory, equivalent to
an infinite matrix theory, that is vastly simpler and completely solvable [12]. However we
3
determined that the γ term is essential for the correct definition of string field theory. Thus
we have isolated the hard part of string field theory in the form of the quadratic “midpoint
correction” term γ. We have shown that all classical solutions, including the true vacuum, of
the interacting string theory, are obtained analytically by first solving the nonlinear equation
explicitly by ignoring γ, and then including the effect of γ in a closed formal expression that
can be evaluated to any order in a perturbative expansion in powers of γ [12].
In our published work so far, we have demonstrated all of the above results explicitly mainly
in the matter sector. We have implied, and sometimes explicitly included, the corresponding
contribution of ghosts either in the bosonic [9] or fermionic [10, 12] version but the details were
not given explicitly. The purpose of the present paper is to provide all the relevant material on
fermionic ghosts which we used previously, in an organized and comprehensive manner. In this
sense, this paper completes the basic formulation of MSFT.
Because we will try to be quite explicit, the content of this paper will be rather technical. The
construction of the Moyal product for fermionic ghosts is basically parallel to the bosonic case
[7, 8, 9], except that we need to be careful in some minor differences, including midpoint issues,
which appear in the ghost fields b, c.
The first point relates to the boundary conditions. While we needed to consider Neumann type
boundary conditions for the matter fields (open strings on the D25 brane), Dirichlet type boundary
conditions appear for the ghost field. Therefore, the Fourier basis is different. The regularization
method developed in [8, 9] was based on the Fourier modes, hence some care is needed to make
the regularization compatible in the matter and ghost sectors. The regularization developed in this
paper can be applied also to the treatment of the matter sector for lower Dp branes (p < 25).
The second point relates to the overlapping conditions of split strings. For the matter fields,
we have to treat only overlapping conditions for the split string degrees of freedom. For the ghost
fields, on the other hand, we should also consider the anti-overlapping conditions. This induces
some changes in the mapping of Witten’s star to Moyal’s star.
The third point is the fermionic nature of Moyal variables. The usual bosonic derivatives which
appear in the definition of the Moyal product get replaced by derivatives of fermionic variables,
and care is needed in the ordering and signs.
The fourth point is the treatment of the midpoint mode. This is a rather delicate issue since,
as in the matter sector, it cannot be determined from the split string formulation. The fermionic
midpoint mode is not part of the Moyal ⋆ product, and it is integrated in the definition of the
action. In the Siegel gauge, the dependence on this extra fermionic variable becomes trivial, and it
drops out in actual computations.
We mention the work of Erler [16] where he defined the Moyal star formulation for the ghost
system mainly in the continuous basis [14].1 There are overlaps of the current paper with his work1There are some works on Moyal structure of Witten’s string field theory.[15]
4
especially in the second and third points mentioned above. The correct treatment of the midpoint
appeared first in our work [10], and his paper was modified subsequently. Beyond the preliminary
level, for correct computation with ghosts, the results of the present paper are needed.
The definition of the fermionic Moyal product outlined above is given in section 2. Because of
its technical nature, we first summarize the fundamental formulae at the beginning of the section,
and explain the full detail in the subsections. Readers who wish to skip the details of the derivation
can proceed to later sections by skipping the latter part. In subsection 2.1, we first discuss the
issue of boundary conditions in general for fermions and the corresponding regularization scheme.
We also give a brief review of [9] in appendix A. Together with it, this paper provides the basic
formulae for both the ghost and matter sectors in a compact form. We then discuss the mapping
from Witten’s star to Moyal’s star in subsection 2.2. Finally we apply these techniques to the
actual bc ghost system in subsection 2.3, and derive the correspondence between the conventional
oscillators and Moyal variables in subsection 2.4.
After this preparation, in section 3, we define the monoid algebra among gaussian string fields,
which provides a useful tool for computations (this is almost a group, except for inverse). We com-
pute the star product of n monoid elements, which as a by-product give the Neumann coefficients
in the ghost sector. These were conjectured in [9] by using a nontrivial relation with the Neumann
coefficients in the matter sector. In this paper, we derive them directly from the fermionic Moyal
product. They satisfy the Gross-Jevicki- nonlinear relations exactly for any frequencies κe, κo and
any N . This fact confirms the consistency of our construction including the midpoint prescription.
As in the matter sector, they are simple functions of the matrix teo for any n-point vertex, and
they can be related to the corresponding matter Neumann coefficients by the simple procedure of
replacing the matrix t by its inverse.
We also give the direct numerical comparison between the analytic form of the Neumann coeffi-
cientsM obtained from conformal field theory and our algebraic expression. We confirmed that the
approximate value for finite N converges to its exact value as N →∞ with the following universal
behavior, Mnm(N)/Mnm(cft) ∼ 1 + anmN−α where the exponent is approximately α ∼ 1.33 for
matter sector and α ∼ 0.67 for the ghost sector for any components of the Neumann coefficients.
While this analysis is of different nature from the other parts in this paper, it is included here since
it gives strong support on the consistency of MSFT in [9] and this paper. It also provides the basis
for numerical computation of MSFT in our future study.
In section 4 we apply the formalism. First, we present the derivation of the open string field
action in the Siegel gauge by including both matter and ghost fields. This action was the starting
point in our recent work [10, 12] where we used the results of the present paper without providing
the details. Finally, we also compute the ghost contribution to the Feynman graphs for the four-
point scattering amplitude for off-shell tachyons, whose matter sector was discussed in [10].
5
2 Moyal’s star from Witten’s star in fermionic ghost sector
In this section we construct the map from Witten’s star product to the Moyal star product for
fermionic variables. The fermionic version of the Moyal ⋆ product was called “anti-Moyal star
product” in the literature [18]. More general star products have also been considered in the context
of deformation quantization of super-Poisson brackets [19]. The anti-Moyal star product will simply
be referred to as the Moyal star product in the following. A first basic construction for the ghost bc
system, following the one in the matter sector [7], was previously discussed in [16], while the correct
treatment of the midpoint was first given in [10]. In this section we give the complete treatment,
including the consistent regularization with the matter sector.
We also define the even basis of ghost modes that is most transparent for our computations,
after defining some other bases as well. We first summarize the main results of this rather lengthy
and technical section:
• The Moyal ⋆ acts on fermionic ghost modes ξ ≡ (xo, po, yo, qo) (o = 1, 3, 5, · · · , 2N − 1) as
(A ⋆ B)(xo, po, yo, qo) = A exp
(θ′
2
∑
o>0
( ←−∂
∂xo
−→∂
∂po+
←−∂
∂yo
−→∂
∂qo+
←−∂
∂po
−→∂
∂xo+
←−∂
∂qo
−→∂
∂yo
))B ,
(2.1)
where θ′ is a parameter which absorbs units, and if desired, could be absorbed away by a
rescaling of the variables which amounts to a choice of units. We note the canonical structure
xo, po′⋆ = θ′δoo′ = yo, qo′ . This odd basis (o) of ghost modes, which was used in [10], is
naturally defined in the process of mapping the Witten star to the Moyal star, including the
treatment of the midpoint. However, a more transparent basis that is more parallel to the
matter sector, which simplifies the overall formalism, is obtained by rewriting the odd basis,
through the following linear canonical transformation, in terms of an even basis xbe, pbe, x
ce, p
ce
(e = 2, 4, · · · , 2N), where the labels b, c refer to the modes of the usual b, c ghosts
xbe := κ−1e
∑
o>0
Seoxo , pbe := κe∑
o>0
Seopo , xce :=∑
o>0
Teoyo , pce :=∑
o>0
qoRoe . (2.2)
This even basis is different than the one defined in [16]. The Moyal product is rewritten as
(A ⋆ B)(xbe, pbe, x
ce, p
ce) = A exp
(θ′
2
∑
e>0
( ←−∂
∂xbe
−→∂
∂pbe+
←−∂
∂xce
−→∂
∂pce+
←−∂
∂pbe
−→∂
∂xbe+
←−∂
∂pce
−→∂
∂xce
))B .
(2.3)
We note the canonical structurexbe, p
be′⋆= θ′δee′ =
xce, p
ce′⋆. The linear transformation
matrices, T,R (and matrices U and vectors v,w which appear in the following) were defined
in [9], while the matrix S appears for the first time in this paper. Their properties are derived
explicitly in subsection 2.1. They play a central role in MSFT since they define a Bogoliubov
transformation from the oscillators in the operator formalism to the Moyal coordinates, and
thus carry essential information about string theory. For instance, see Eqs.(2.4,2.6) in the
6
next paragraph. For the moment, we just mention that they satisfy the inverse properties
TR = RT = SS = SS = 1 which prove that the two versions of the star product (2.1) and
(2.3) are canonically equivalent. Throughout this paper, the bar ( ) means the transpose of
matrices or vectors. The relation of split strings to the Moyal star is discussed in subsection
2.2.
• In the operator formulation, the ghost sector of the string field |Ψ〉 is represented in the Fock
space of the bc ghost oscillators. The transformation to the Moyal field A (ξ) as a function
of noncommutative coordinates ξ = (x, p) is obtained through the Fourier transformation [7]
of the coordinate representation of |Ψ〉. The whole procedure is more neatly expressed, as in
the matter sector [9], by the inner product with a particular bra state 〈ξ0, ξ1, ξ2| in the Fock
space, where ξ1 = (xo, po) , ξ2 = (yo, qo) are the noncommutative fermionic coordinates, and
ξ0 is a fermionic variable related to the zero mode dependence of |Ψ〉
〈ξ0, ξ1, ξ2| = −2−2N (1 + ww)−14 〈Ω|c−1e
−ξ0(c0−√2wce)ecebe−cobo−2iξ1M
(o)0 ξ2−ξ1λ1−ξ2λ2 , (2.4)
A(ξ0, ξ1, ξ2) = 〈ξ0, ξ1, ξ2|Ψ〉 . (2.5)
Here 〈Ω| is the SL(2, R) invariant bra Fock vacuum, bn, cn are the conventional ghost oscilla-
tors, and the matrices M(o)0 and λ are defined as
M(o)0 =
(121o 0
0 2θ′2(SR)oo′
), λ1 =
(−i√2co
2iθ′ Soe
(−√2be + weξ0
)), λ2 =
(−√2bo
−2√2
θ′ Roece
).
(2.6)
The matrices we, Seo, Roe are functions of the oscillator frequencies κe, κo as given below.
Through Eqs.(2.1,2.4) we map Witten’s star (⋆W ) into the Moyal’s star ⋆ as follows
〈ξ0, ξ|Ψ1 ⋆W Ψ2〉 ∼ 〈ξ0, ξ|Ψ1〉 ⋆ 〈ξ0, ξ|Ψ2〉 , (2.7)
|Ψ1 ⋆W Ψ2〉3 = 1〈Ψ1|2〈Ψ2|V3〉123, 1〈Ψ1| = 14〈V2|Ψ1〉4, 2〈Ψ2| = 25〈V2|Ψ2〉5 . (2.8)
We note that the product is local in ξ0, while ξ0 plays a similar role to the midpoint coordinate
xµ in the matter sector. The Moyal star reproduces correctly the three string vertex |V3〉 andthe reflector |V2〉 of the operator formalism [2]. The details are given in subsection 2.3. The
precise correspondence including the zero mode is in section 3.
• Eq.(2.4) is enough to derive the connection between the conventional operator formalism
and the Moyal star formalism. For example, the action of the standard oscillators on the
Fock space field |Ψ〉 can be rewritten in terms of their Moyal images acting on the field
A = 〈ξ0, ξ1, ξ2|Ψ〉 through the star product, as follows
b0|Ψ〉 ↔ −ξ0A ,
c0|Ψ〉 ↔(− ∂
∂ξ0+θ′
2v∂
∂qo
)A ,
bo|Ψ〉 ↔1√2
(βbo ⋆ A− (−1)|A|A ⋆ βb−o
),
7
co|Ψ〉 ↔1√2
(βco ⋆ A+ (−1)|A|A ⋆ βc−o
), (2.9)
be|Ψ〉 ↔1√2
(βbe ⋆ A+ (−1)|A|A ⋆ βb−e
)+ w′
e ξ0A ,
ce|Ψ〉 ↔1√2
(βce ⋆ A− (−1)|A|A ⋆ βc−e
),
where βbo, βco are fields in Moyal space which obey oscillator relations under the ⋆ product
βbo :=1
2
(2
θ′q|o| − iǫ(o)x|o|
), βco :=
1
2
(y|o| − i
2
θ′ǫ(o)p|o|
), βbo, βco′⋆ = δo+o′ . (2.10)
On the other hand βbe, βce are not independent from the βbo, β
co. Rather, they are their Bogoli-
ubov transforms which obey the following relations
βbe =∑
o
βbo U−1−o,e , β
ce =
∑
o
Ue,−oβco , βbe, βce′⋆ = δe+e′ , (2.11)
andβb−o, β
ce
⋆= Ue,−o where the matrix Ue,−o will be given below. With these formulas,
one can directly translate operators in Fock space into their images which act in Moyal space.
The proof of these formulas and some variants are discussed in subsection 2.4. In this way
we derive the explicit form of the Virasoro operator L0 which acts in Moyal space
L0 =
2N∑
k=1
κk(βb−kβ
ck + βc−kβ
bk) (2.12)
=
2N∑
k=1
κk + i∑
o>0
κo
(xoyo +
∂
∂xo
∂
∂yo
)+ i∑
o>0
κo
(4
θ′2poqo +
θ′2
4
∂
∂po
∂
∂qo
)
+4i
θ′2(1 + ww)
(∑
o>0
κovopo
)(∑
o′>0
vo′qo′
)+
2i
θ′(1 + ww)
(∑
o>0
voκopo
)ξ0 .
This was used in [10, 12]. Here βbk, βck are not the Moyal fields βbo, β
co or βbe, β
ce given in
Eqs.(2.10,2.11); rather, they are differential operators that obey the standard oscillator rela-
tions, and which are derived from the star products of the fields βbo, βco or βbe, β
ce as will be
shown below.
2.1 Half string formalism and regularization
We start from full string functions ψ(σ), 0 ≤ σ ≤ π with Dirichlet boundary conditions at σ = 0, π,
and discuss their split string formulation in the interval 0 ≤ σ ≤ π/2. Such functions have a
Fourier expansion with only sine modes in the full string formalism. By contrast, the corresponding
problem in the matter sector involved only cosine modes because of Neumann boundary conditions
at σ = 0, π. We collect the basic formulae in the appendix A.1. The essential step was the
construction of the regularization [8, 9] which is needed to avoid the associativity anomaly. We
8
give a regularization of the ghost sector for the sine mode expansion which is compatible with the
previous results.
A full string function ψ(σ) which satisfies Dirichlet boundary conditions ψ(0) = ψ(π) = 0 is
expanded as
ψ(σ) =√2
∞∑
n=1
ψn sinnσ , ψn =
√2
π
∫ π
0dσψ(σ) sin nσ . (2.13)
We decompose such a field ψ(σ) into left half l(σ) and right half r(σ) as follows
ψ(σ) =
l(σ) 0 ≤ σ ≤ π
2
r(π − σ) π2 ≤ σ ≤ π
, ψn =
√2
π
∫ π2
0dσ(l(σ) − (−1)nr(σ)) sinnσ . (2.14)
As we have seen in [7, 8], we have some arbitrariness in the choice of the boundary condition for
the split string functions l(σ), r(σ) at the midpoint σ = π/2. They can be expanded using either
odd or even modes according to the two possible choices of the boundary condition at the midpoint
σ = π2 as discussed below.
2.1.1 Dirichlet at end point, Neumann at midpoint (DN)
First we consider Neumann boundary conditions at the midpoint σ = π/2, while we have Dirichlet
boundary conditions at the end point for the split string functions l(σ), r(σ):
l(0) = r(0) = 0 , l′(π/2) = r′(π/2) = 0 . (2.15)
We can expand them using odd sine modes o = 1, 3, 5, · · ·
l(σ) =√2
∞∑
o=1
lo sin oσ , lo =2√2
π
∫ π2
0dσ l(σ) sin oσ , (2.16)
r(σ) =√2
∞∑
o=1
ro sin oσ , ro =2√2
π
∫ π2
0dσ r(σ) sin oσ . (2.17)
By comparing the mode expansions Eqs.(2.13,2.16,2.17) with (2.14), we obtain the correspondence
between the full and split string variables,
lo = ψo + Soeψe , ro = ψo − Soeψe , (2.18)
or the inverse
ψe =1
2Seo(lo − ro) , ψo =
1
2(lo + ro), (2.19)
where e = 2, 4, 6, · · · , and the matrix Seo is given by
Seo =4
π
∫ π2
0dσ sin eσ sin oσ =
4io−e+1e
π(e2 − o2) . (2.20)
The above mappings are consistent because Seo is an orthogonal matrix:
SS = SS = 1 . (2.21)
9
The continuity condition at the midpoint ψ(π/2) = l(π/2) = r(π/2) is
∞∑
o=1
woψo =∞∑
o=1
wolo =∞∑
n=1
woro (2.22)
where we defined the odd vector w associated with the midpoint
wo =√2 sin
oπ
2=√2 io−1 . (2.23)
Eq.(2.22) holds thanks to the identity (Sw)e =∑∞
o=1 Seowo = 0. This equation implies that S has a
singular eigenvector even though it has an inverse, which is just its transpose S, as stated in (2.21).
This esoteric relation is possible because S is an infinite matrix. However it causes an associativity
anomaly with respect to matrix products, which in turn feeds into associativity anomaly of string
field star products, as discussed in [8].
An example of the associativity anomaly is S(Sw) = 0, but (SS)w = w. Each single sum
indicated by the parentheses has a unique answer, but the double sums are ambiguous. The
reason is that, due to infinite sums, in the first expression there are terms of the form ∞/∞which are ambiguous. Since these matrices appear in many physical computations we must give
an unambiguous definition of the matrix product. We will resolve this ambiguity in computations
successfully by a regularization procedure.
2.1.2 Dirichlet at end point, Dirichlet at midpoint (DD)
We consider another possibility: we define the midpoint of the string ψ := ψ(π/2), and impose
Dirichlet boundary conditions at both σ = 0, π/2 on the split string functions l(σ), r(σ),
l(0) = r(0) = 0 , l(π/2) = r(π/2) = ψ. (2.24)
We note that the midpoint value ψ is an additional degree of freedom in the split string basis and
cannot be chosen arbitrarily. We expand l(σ), r(σ) using even sine modes, e = 2, 4, 6, · · ·
l(σ) =2
πσψ +
√2
∞∑
e=2
le sin eσ , le =2√2
π
∫ π2
0dσ
(l(σ)− 2
πσψ
)sin eσ , (2.25)
r(σ) =2
πσψ +
√2
∞∑
e=2
re sin eσ , re =2√2
π
∫ π2
0dσ
(r(σ)− 2
πσψ
)sin eσ . (2.26)
Again, by comparing the mode expansions Eqs.(2.13,2.25,2.26) with (2.14), the correspondence
between split and full string variables is obtained
ψ = woψo , le = ψe + Teoψo , re = −ψe + Teoψo , (2.27)
or the inverse
ψe =1
2(le − re) , ψo = uoψ +
1
2Soe(le + re) , (2.28)
10
where
Teo =4o2 io−e+1
πe(e2 − o2) = Seo + vewo =1
e2Seoo
2, (2.29)
uo =4√2
π2
∫ π2
0dσσ sin oσ =
4√2 io−1
π2o2, (2.30)
ve = −4√2
π2
∫ π2
0dσσ sin eσ =
2√2 ie
eπ. (2.31)
The maps (ψ, le, re)↔ (ψn) are consistent by the relation
uowo = 1, T S = 1 , T u = 0, Sw = 0 . (2.32)
We can prove the following relations among infinite matrices S, T and vectors u, v, w by straight-
forward computation:
S = κ2e T κ−2o , S = T − v ¯w , Sv = −u , Su = −v , ¯T v = −u+
1
3w , (2.33)
SS = SS = T S = 1, ST = 1− u ¯w , Sw = T u = 0 , ¯wu = 1 , ¯vv = ¯uu =1
3, (2.34)
where κe, κo are the diagonal matrices κe = diag(2, 4, 6 · · · ) and κo = diag(1, 3, 5, · · · ). This algebrais similar to the one among the infinite matrices T,R, v, w which appeared in the matter sector [8],
where a full string function ψ(σ) is expanded in terms of cosine modes (see §A.1).
2.1.3 Regularization
In the split string formulation given in §A.1.1 §A.1.2 §2.1.1 §2.1.2, we encountered a set of infinite
dimensional matrices T,R, S, T and vectors w, v, w, v, u. These represent Bogoliubov transforma-
tions between odd and even modes, with (T,R,w, v) appearing when the full string is expanded
in terms of cosine modes, and (S, T , w, v, u) appearing when the full string is expanded in terms
of sine modes. Such transformations are essential in the Moyal formulation since they carry basic
information about string theory. We note that, the Moyal star product itself, which is applied
independently for each mode, has no specific information about string theory, and as such is a more
general structure.
In the analysis of the matter sector [8] as well as the sine mode expansion given so far, there
appears an associativity anomaly in the matrix algebra of these matrices. This originates from
the infinite dimensionality of these matrices. It produces ambiguities in computations in string
field theory. To have a well defined theory, it is mandatory to define a deformed, unambiguous,
associative algebra that preserves the basic matrix algebraic structure of these matrices [8, 9]. Such
a deformation contains a parameter N, that corresponds to the rank of the matrices. The original
definition of these matrices given above is reproduced by taking the limit N →∞ of this parameter.
All computations in the open string field theory are performed unambiguously with finite N, and
11
the correct value in string theory is obtained at the end of the computation by taking the limit
N →∞. This is the basic strategy for practical computations in the MSFT proposal.
The deformed set of matrices for T,R,w, v that have correctly reproduced string theory was
proposed in [8, 9]. Since an additional set of matrices S, T , w, v, u have appeared in the ghost sector
we need to obtain their deformation consistently with the matter sector. For that purpose, we start
from the infinite dimensional matrix U and vectors w′, v′ defined in [9],
U−e,o =2
π
io−e−1
o− e , U−1−o,e =
2
π
e
o
io−e−1
o− e , w′e = i−e+2 , v′o =
2
π
io−1
o, (2.35)
where e (o) now run over both positive and negative integers ±2,±4, · · · (resp. ±1,±3, · · · ). In
single sums these matrices satisfy
∑
o
U−e,oU−1−o,e′ = δe,e′ ,
∑
e 6=0
U−1−o,eU−e,o′ = δo,o′ , (2.36)
which we denote UU−1 = 1e, U−1U = 1o for short in the following. More importantly, there exists
the following matrix relations among them,
U−1 = κ′o−1Uκ′e , U−1 = U + v′w′ , v′ = Uw′ , w′ = U−1v′ , (2.37)
where κ′e = diag(· · · ,−4,−2, 2, 4, · · · ), and κ′o = diag(· · · ,−3,−1, 1, 3, · · · ) are the diagonal ma-
trices which specify the spectrum. These relations will be used as the defining relations. From
them it is possible to derive the matrices themselves, as given in Eq.(2.35). Therefore, they will
be used as the basic relations that are also satisfied by the deformed matrices, as given below.
The first relation implies that U defines an invertible Bogoliubov transformation between even and
odd spectra (see also Eq.(2.11)). The second relation shows that the transformation U is almost
orthogonal except for the vectors v′, w′ which are associated with the midpoint mode. Finally the
last two define the relation between the vectors.
On the other hand, the matrices (2.35) also satisfy
UU = 1e, Uv′ = 0, v′v′ = 1 , (2.38)
which break the associativity [8, 9]. Therefore, these relations will be deformed in the regulated
theory, as seen below. Of course these equations will hold when the regulator is removed.
All the other matrices are written in terms of U,w′, v′, (for e, o > 0),
Teo = U−e,o + Ue,o , Roe = U−1−o,e + U−1
o,e , we =√2w′
e, vo =√2v′o , (2.39)
Seo = U−e,o − Ue,o = U−1−o,e − U−1
o,e , Teo = κ−1e Tκo , (2.40)
u =2
πκ−1o v , w =
π
2κov , v = − 2
πκ−1e w , (2.41)
where κo and κe are restrictions of κ′o and κ′e to the positive sector. These definitions in terms of
U,w′, v′ together with the relations (2.37) among U,w′, v′ are sufficient to derive all the relations
12
among T,R,w, v, S, T , u, v, w. Therefore, we may use these relations as the definitions of these
matrices and vectors even when we use the regularization of U,w′, v′.
In [8, 9, 10], the regularization of U, v′, w′ is given explicitly. We truncate the size of U, v′, w′
to 2N while keeping their property of Bogoliubov transformation between even and odd spectrum.
It turns out that one may take the even and odd frequencies κe, κo as arbitrary functions of the
positive integers (e, o) , while keeping the reflection property of κ′e,o to extend the definition to
negative integers κ′−e = −κ′e, κ′−o = −κ′o. Therefore we put
κ′e = ǫ(e)κ|e| , κ′o = ǫ(o)κ|o| . (2.42)
We suppose implicitly that κ′e 6= 0, κ′o 6= 0 and that these are not degenerate.
The matrices U,U−1 and vectors w′, v′ can then be derived from the defining relations (2.37) as
functions of the arbitrary spectral parameters κe, κo as (see appendix B for details of the derivation)
U−e,o =w′ev
′oκ
′o
κ′e − κ′o, U−1
−o,e =w′ev
′oκ
′e
κ′e − κ′o, U−e,o = Ue,−o , U−1
−o,e = U−1o,−e , (2.43)
√2w′
e = w|e| = i2−e
∏o′>0
∣∣∣κ2|e|/κ2o′ − 1∣∣∣12
∏e′>0,e′ 6=|e|
∣∣∣κ2|e|/κ2e′ − 1∣∣∣12
, w′e = w′
−e , (2.44)
√2 v′o = v|o| = i|o|−1
∏e′>0
∣∣∣1− κ2|o|/κ2e′∣∣∣12
∏o′>0,o′ 6=|o|
∣∣∣1− κ2|o|/κ2o′∣∣∣12
, v′o = v′−o (2.45)
where now the indices (e, o) run over the finite set e = ±2,±4, . . . ,±2N and o = ±1,±3, . . . ,±(2N−1). It is easy to check explicitly that in the limit N → ∞ and κe = e, κo = o, these expressions
reduce to Eq.(2.35).
The regulated expressions for these matrices look considerably more complicated than their
large N limit. Therefore, it may appear that this would create a problem in analytic computations.
Actually this is not the case at all, because in analytic computations one uses the matrix relations
satisfied by these matrices rather than the explicit matrices themselves. The relations are preserved
in the regularized version for any N, and they look the same as their N =∞ counterpart. Therefore
analytically the expressions in any computation look the same in the regulated or infinite versions
as long as they are written in terms of these matrices without using their explicit form. The
explicit construction of the regulated version insures associativity and eliminates the ambiguity of
the associativity anomaly as explained below. Thus, in addition to the basic defining relations (2.37)
which are the same for any N, including N =∞, there are more relations among U,U−1, v′, w′ that
can now be derived from the defining relations alone for any N,κ′e, κ′o :
UU−1 = 1 , U−1U = 1 , U−1U−1 = 1 + w′w′ , UU = 1− v′v′ , (2.46)
UU = 1− w′w′
1 + w′w′ , Uv′ =w′
1 + w′w′ , v′v′ =w′w′
1 + w′w′ , (2.47)
13
U−1w′ = v′(1 + w′w′) , U−1U−1 = 1 + v′v′(1 + w′w′) , (2.48)
1 + w′w′ =
∏e κ
′e∏
o κ′o
=
∏e>0 κ
2e∏
o>0 κ2o
. (2.49)
In particular note that Eqs.(2.38) are now deformed into Eqs.(2.47), and that w′w′ → 2N → ∞in the large N limit. Thus, the deformed algebra actually holds also at N = ∞; it simply makes
explicit the behavior as a function of N.With this, the associativity anomaly hidden in the original
algebra is now resolved and the matrix algebra for all the matrices U,U−1, T,R, S, T , v, w, v, w, u
defined through Eqs.(2.39–2.45) becomes associative.
In particular, from Eq.(2.39–2.41) we obtain the regularized matrices, such as T,R, S,
Teo =wevoκ
2o
κ2e − κ2o, Roe =
wevoκ2e
κ2e − κ2o, Seo =
wevoκeκoκ2e − κ2o
. (2.50)
From the relations among U,w′, v′ we can derive the relations among T,R, v, w, S, T , u, v, w. The
deformed algebra among T,R,w, v (A.18) is already given in [8], while the deformed algebra among
S, T , u, v, w which replaces (2.33,2.34) is2,
S = κ2eT κ−2o , S = T − v ¯w , Sv = −u , Su = −v , (2.51)
SS = 1 , SS = 1 , ST = 1− u ¯w , T S = 1− κ−1e wwκe1 + ww
, (2.52)
Sw =2
π
κew
1 + ww, T u =
2
π
κ−1e w
1 + ww, ¯T v = −u+
(2
π
)2
wκ−2e ww , (2.53)
¯wu = vv =ww
1 + ww, ¯vv =
(2
π
)2
wκ−2e w , ¯uu =
(2
π
)2
vκ−2o v . (2.54)
Furthermore, note that for any N,κe,κo we have
wκ−2e w = vκ−2
o v =∑
o>0
κ−2o −
∑
e>0
κ−2e . (2.55)
The right hand side converges to π2
12 if we take the open string limit κe = e, κo = o, N =∞. More
relations of this type can be found in the next subsection.
2.1.4 GL(N |N) supergroup property of the regulator
In this subsection we take a small detour to make an observation on the regulator whose significance
for computations in MSFT is not yet fully apparent, but which is mathematically interesting, and
could be useful in future applications. Many computations in MSFT boil down to expressions
of the form wf (κ)w where f (κ) is a matrix constructed from the frequencies κe, κo through the
regulated matrices we discussed above. Therefore, we are interested in developing analytic methods
of computation involving such expressions, in particular for arbitrary frequencies κe, κo. In such
2Note that for finite N , the continuity condition at the midpoint (2.22) is not satisfied because Sw 6= 0. However,
we recover it, as well as all other infinite matrix relations by taking the open string limit κe = e, κo = o,N = ∞.
14
computations the properties of the supergroup GL(N |N) mysteriously makes an appearance, as
follows.
By using explicitly the expression for we given in Eq.(2.44), we have
wf(κ2e)w =
∑
e
f(κ2e) deto
(κ2eκ
−2o − 1
)
dete′ 6=e
(κ2eκ
−2e′ − 1
) =
∮dz
2πi
f (z)
z
det(1− zκ−2
o
)
det(1− zκ−2
e
) , (2.56)
where the contour encircles only the poles at z = κ2e. The contour may then be deformed to evaluate
the integral. When f(κ2e)= 1 or κ−2
e the results have already been given in Eqs.(2.49,2.55). We
note that these may be written in the form
ww = −1 + Sdet(κ2), wκ−2
e w = −Str(κ−2
), (2.57)
where we used the superdeterminant (Sdet) and supertrace (Str) by treating the matrix
κ =
(κe 0
0 κo
)(2.58)
as if it is a graded GL(N |N) super matrix. When f(κ2e)= κ−2n
e , with n = 1, 2, · · · , we note
that the contour integral is precisely the integral representation of the supercharacter of GL(N |N)
for the representation of GL(N |N) described by a Young supertableau with a single row with n
superboxes [17]
wκ−2ne w = −χn
(κ−2
). (2.59)
The expression for the supercharacter χn (M) for any supermatrixM, can also be written in terms
of supertraces of powers of M in the fundamental representation, as given in [17]. By taking
advantage of this observation we evaluate χn
(κ−2
)in terms of the supertraces of powers of κ−2.
For example, χ2 (M) = 12Str
(M2)+ 1
2 (Str (M))2 , which gives the following interesting expression
for the sum
wκ−4e w = −χ2
(κ−2
)= −1
2
[Tr(κ−4e
)− Tr
(κ−4o
)]− 1
2
[Tr(κ−2e
)− Tr
(κ−2o
)]2. (2.60)
We can check the correctness of Eq.(2.59) in the limit κe = e, κo = o, N = ∞. In this limit,
since we →√2 i2−e, we can evaluate the sum on the left side directly in terms of the zeta func-
tion, wκ−2ne w → 2
∑∞k=1 (2k)
−2n = 222nζ (2n) , and then compare it to the value generated by the
supercharacter −χn
(κ−2
)in the limit.
For example, for n = 1 the right hand side of Eq.(2.59) is already given following Eq.(2.55),
and this agrees with the left hand side which is 222ζ (2). Similarly, for n = 2 the left hand side of
Eq.(2.60) gives 224 ζ (4) =
1720π
4 while the right hand side gives 716ζ (4) − 1
8 (ζ (2))2 = 1
720π4 which
agree. The GL(N |N) supergroup property of these sums is intriguing. It may be the signal of an
underlying mathematical structure that could be helpful in computations in MSFT.
15
2.2 Moyal ⋆ from Witten’s ∗ for fermionic modes
The second step in constructing the map from Witten’s star to Moyal’s star is to perform the
Fourier transformation from position space to momentum space for a subset of string modes [7].
We recall the definition of Witten’s ∗-product for functions of split strings in the ghost sector
Ψ1 ∗Ψ2[l(σ), r(σ)] ∼∫Dz(σ)Ψ1[l(σ), z(σ)]Ψ2[±z(σ), r(σ)] . (2.61)
For the bc ghost sector, we consider two types of overlapping conditions [2]. The anti-overlapping
condition (resp. overlapping condition) is defined by choosing the minus (resp. plus) sign in this
formula. In both cases the Witten ∗ product is mapped to the Moyal ⋆ product as follows.
We denote the string field as Ψ[l, r] in the split string formulation and as Ψ[x, p] in the Moyal
formulation. The variables l, r, x, p are all fermionic and we consider the simplified situation where
each of them represents a single degree of freedom. The generalization to multiple variables is
straightforward. For the simplified setup we define Witten’s star product in the split string formal-
ism (ignoring the midpoint for the time being) as
Ψ1 ∗Ψ2[l, r] = (−1)|Ψ1|∫dwΨ1[l, w] Ψ2[±w, r] . (2.62)
The sign factor (−1)|Ψ1| (Grassmann parity of Ψ1) is needed to make the ∗ product associative.
We define the mapping from a string field in the split string picture Ψ[l, r] to the Moyal picture
Ψ[x, p] by using the Fourier transform3
Ψ[x, p] = ±∫dy e−py Ψ
[±x+
y
2, x∓ y
2
], (2.63)
Ψ[l, r] = ±∫dp ep(l∓r) Ψ
[r ± l2
, p
]. (2.64)
Witten’s star for Ψ is then mapped to Moyal’s star for Ψ:
Ψ1 ⋆ Ψ2[x, p] = Ψ1[x, p] exp
(∓1
2
(←−∂
∂x
−→∂
∂p+
←−∂
∂p
−→∂
∂x
))Ψ2[x, p] . (2.65)
The derivation of this correspondence is completely parallel to the bosonic case [7]:
Ψ1[x, p]e∓ 1
2
( ←−∂∂x
−→∂∂p
+←−∂∂p
−→∂∂x
)
Ψ2[x, p]
=
(±∫dy1 e
−py1 Ψ1
[±x+
y12, x∓ y1
2
])e∓ 1
2
( ←−∂∂x
−→∂∂p
+←−∂∂p
−→∂∂x
) (±∫dy2 e
−py2 Ψ2
[±x+
y22, x∓ y2
2
])
= (−1)|Ψ1|∫dy1dy2
(Ψ1
[±x+
y12, x∓ y1
2
]e−py1
)e∓ 1
2
(←−∂∂x
−→∂∂p
+←−∂∂p
−→∂∂x
) (e−py2 Ψ2
[±x+
y22, x∓ y2
2
])
= (−1)|Ψ1|∫dy1dy2Ψ1
[±x+
y12, x∓ y1
2
]e±
12
←−∂∂x
y2e−p(y1+y2)e∓12y1−→∂∂xΨ2
[±x+ y2
2, x∓ y2
2
]
3If l and r consist of N variables, the sign factor on the right hand sides of Eqs.(2.62)(2.63)(2.64) become
(−1)|Ψ1|N , (±1)N , (±1)N respectively. In particular, they are trivial in the case that N is even.
16
= (−1)|Ψ1|∫dy1dy2 e
−p(y1+y2)Ψ1
[±x+
y1 + y22
, x∓ y1 − y22
]Ψ2
[±x− y1 − y2
2, x∓ y1 + y2
2
]
= ±(−1)|Ψ1|∫dye−py
∫dzΨ1
[±x+
y
2, z]Ψ2
[±z, x∓ y
2
]
= ±∫dye−py (Ψ1 ∗Ψ2)
[±x+
y
2, x∓ y
2
]=: Ψ1 ⋆ Ψ2[x, p] . (2.66)
The form of the product in Eq.(2.65) is similar to the ordinary Moyal product although the deriva-
tives in the exponential are for fermionic variables. This Moyal ⋆ product is associative and non-
commutative.
From Eq.(2.64), we also obtain the correspondence between the definition of trace in the split-
string formulation, which we take with the anti-periodic condition, and that of Moyal one which is
given by an integration in “phase space”
TrΨ :=
∫dzΨ[±z, z] = ±
∫dxdp Ψ[x, p] =: ±Tr Ψ . (2.67)
2.3 Moyal ⋆ product in the bc ghost sector
In this section we define the Moyal ⋆ product which represents Witten’s star product using the
results in §2.1, §2.2, §A.1. We first review the conventional operator formalism to fix the notation
in the bc ghost sector. We take the ghost coordinates b(σ), c(σ) and their conjugates πb(σ), πc(σ)
b±(σ) =∞∑
n=−∞bne
±inσ = πc(σ)∓ ib(σ) , c±(σ) =∞∑
n=−∞cne
±inσ = c(σ) ± iπb(σ) . (2.68)
We introduce the fermionic variables xn, yn and their conjugates pn, qn as follows
b(σ) = −∞∑
n=1
(bn − b−n) sin nσ = i√2
∞∑
n=1
xn sinnσ , (2.69)
c(σ) = c0 +
∞∑
n=1
(cn + c−n) cos nσ = c0 +√2
∞∑
n=1
yn cosnσ , (2.70)
πb(σ) =∞∑
n=1
(cn − c−n) sin nσ = −i√2
∞∑
n=1
pn sinnσ , (2.71)
πc(σ) = b0 +
∞∑
n=1
(bn + b−n) cosnσ = b0 +√2
∞∑
n=1
qn cosnσ . (2.72)
The nonzero modes xn, yn, pn, qn are related to bn, cn:
xn =i√2(bn − b−n) , yn =
1√2(cn + c−n) , pn =
i√2(cn − c−n) , qn =
1√2(bn + b−n) , (2.73)
and the canonical commutation relation bn, cm = δm+n,0 can be rewritten as
xn, pm = δn,m , yn, qm = δn,m , n,m = 1, 2, · · · . (2.74)
17
We represent the string field by treating c0 and xn, yn as the “position” coordinates. The translation
between Fock space representation and position representation is made through
Ψ(c0, xn, yn) = 〈c0, xn, yn|Ψ〉 . (2.75)
Here we introduced the bra state 〈c0, xn, yn| (and the corresponding ket state) as states in Fock
space which satisfy the eigenvalue conditions for the operators c0, xn, yn,
〈c0, xn, yn|c0 = 〈c0, xn, yn|c0 , 〈c0, xn, yn|xn = 〈c0, xn, yn|xn , 〈c0, xn, yn|yn = 〈c0, xn, yn|yn ,c0|c0, xn, yn〉 = c0|c0, xn, yn〉 , xn|c0, xn, yn〉 = xn|c0, xn, yn〉 , yn|c0, xn, yn〉 = yn|c0, xn, yn〉 .
Explicitly these are given by
〈c0, xn, yn| = 〈Ω|c−1c0 exp
(c0b0 +
∞∑
n=1
(−cnbn − i
√2cnxn +
√2ynbn + iynxn
)), (2.76)
|c0, xn, yn〉 = exp
(b0c0 +
∞∑
n=1
(−b−nc−n + i
√2xnc−n +
√2b−nyn − ixnyn
))c0c1|Ω〉 (2.77)
where 〈Ω|, |Ω〉 represents the conformal vacuum4 normalized as 〈Ω|c−1c0c1|Ω〉 = 1. These bras and
kets satisfy the normalization and completeness relations
〈c0, xn, yn|c′0, x′n, y′n〉 = −(c0 − c′0)∞∏
n=1
(−2i(xn − x′n)(yn − y′n)
), (2.78)
−∫dc0
∫ ∞∏
n=1
dxndyn2i
|c0, xn, yn〉〈c0, xn, yn| = 1 . (2.79)
Witten’s star product for the ghost sector is defined by the (anti-)overlapping conditions [2],
b±(r)(σ)− b±(r−1)(π − σ) = 0 , c±(r)(σ) + c±(r−1)(π − σ) = 0 , (2.80)
for r = 1, 2, 3mod 3 , σ ∈ [0, π/2], or equivalently
b(r)(σ)− b(r−1)(π − σ) = 0 , c(r)(σ) + c(r−1)(π − σ) = 0 , (2.81)
π(r)b (σ) + π
(r−1)b (π − σ) = 0 , π(r)c (σ) − π(r−1)
c (π − σ) = 0 . (2.82)
These (anti-)overlapping conditions for bc ghost will be used to define the mapping from Witten’s
∗ to Moyal’s ⋆ by using Eq.(2.63) defined in the previous section.
To apply the formulation in §2.2, we need to specify the boundary conditions of the split string
variables, since we have to use the Bogoliubov transformation given in §2.1/§A.1 accordingly. At
σ = 0, π, b(σ) (resp. c(σ)) satisfies the Dirichlet (resp. Neumann) boundary condition. On the
other hand, at the midpoint, there are two options, namely Neumann or Dirichlet type boundary
4We take the convention that |Ω〉 is Grassmann even and 〈Ω| is odd.
18
conditions. In the following we choose the Neumann condition for b(σ) and Dirichlet condition for
c(σ)5.
In the split string language, the left and right halves of b(σ), lb(σ), rb(σ), satisfy Dirichlet at
σ = 0 and Neumann at σ = π/2, while the left and right halves of c(σ), lc(σ), rc(σ), satisfy Neumann
at σ = 0 and Dirichlet at π/2. With this choice, lb(σ), rb(σ) are expanded by using odd sine modes:
sin oσ, o = 1, 3, 5, · · · , and lc(σ), rc(σ) by using odd cosine modes: cos oσ, o = 1, 3, 5, · · · :
lb(σ) = i√2
∞∑
o=1
lbo sin oσ , rb(σ) = i√2
∞∑
o=1
rbo sin oσ , (2.83)
lc(σ) = c+√2
∞∑
o=1
lco cos oσ , rc(σ) = c+√2
∞∑
o=1
rco cos oσ . (2.84)
From Eqs.(2.69)(2.70)(2.18)(A.7), we have the relations between split- and full-string variables
lbo = Sxe + xo, rbo = −Sxe + xo , (2.85)
c = c0 − wye, lco = Rye + yo, rco = Rye − yo (2.86)
where we used a matrix notation. Witten’s ∗ product for the split string formulation is written as6
A ∗ B(c, lbo, lco, r
bo, r
co) =
∫ ∏
o>0
(idηbodη
co
)A(c, lbo, l
co, η
bo, η
co)B(c, ηbo,−ηco, rbo, rco) . (2.87)
The string field in the split-string formulation is identified with the usual position representation
Ψ(x (σ)), which is written in terms of modes
A(c, lbo, lco, r
bo, r
co) ∼ Ψ(c0, xn, yn) := 〈c0, xn, yn|Ψ〉 . (2.88)
In order to map it to the Moyal formulation, we compare Eqs.(2.85)(2.86) with Eq.(2.63), and note
the similarities (we add prime ′ to distinguish the variables with anti-overlapping condition from
the variables with overlapping conditions)
Sxe + xo ∼ x+y
2, −Sxe + xo ∼ x−
y
2, Rye + yo ∼ −x′ +
y′
2, Rye − yo ∼ x′ +
y′
2(2.89)
or equivalently7
xo ∼ x , Sxe ∼y
2, yo ∼ −x′ , Rye ∼
y′
2. (2.90)
5The other choice (Neumann for b and Dirichlet for c at the midpoint) is discussed in the appendix D. It gives
equivalent but more complicated expression for the Moyal formulation.6Here we consider naive overlapping condition. To obtain the conventional Witten’s star product, as we will show,
we should treat midpoint variable c more carefully. The phase factor i in the measure is only convention so that it
is “real” (idxedye)† = idxedye. Here we define complex conjugate for fermionic variables ξ, ξ′ as (ξξ′)† = (ξ′)†(ξ)† .
7We note that the odd modes correspond to the “x-variable” of phase space in the Moyal formulation of ghosts.
By contrast, in the matter sector, the even modes played the corresponding role (see Eq.(30) in [7]).
19
Thus, using Eqs.(2.63)(2.88), we obtain the map from the field in the position representation to
the Moyal representation
A(c, xo, po, yo, qo) := 2−2N (1 + ww)−14
∫ ∏
e>0
(i−1dxedye
)e−2poSxe−2qoRyeΨ(c+ wye, xn, yn) . (2.91)
At this point, we used the MSFT regularization scheme, by truncating the ghost modes xn, yn
to n ≤ 2N , and using the parameters (N,κe, κo) given in the previous section. Thus, w,R, S
are redefined in Eqs.(A.17)(A.16)(2.50) and 22N , ww are finite. We fixed the normalization factor(det(16SR)
)− 12 = 2−2N (1 + ww)−
14 consistently with the trace that will be given later. Hence,
from Eqs.(2.65)(2.90), the Moyal ⋆ product that corresponds to Witten’s ∗ product is
A ⋆ B(c, xo, po, yo, qo)
= A(c, xo, po, yo, qo) e− 1
2
∑o>0
( ←−∂
∂xo
−→∂
∂po+←−∂
∂yo
−→∂
∂qo+←−∂
∂po
−→∂
∂xo+←−∂
∂qo
−→∂
∂yo
)
B(c, xo, po, yo, qo) . (2.92)
We introduced an arbitrary parameter θ′ which is the analog of θ in the matter sector[9] to absorb
units. We note that the product is local as a function of the midpoint c, while c is related to the
center or mass variable c0 by
c0 = c+ wye . (2.93)
By rescaling variables and performing Fourier transformation with respect to c, we arrive at the
definition of the string field in MSFT
A(ξ0, xo, po, yo, qo)
=
∫dce−ξ0cA(c, xo,−po/θ′, yo,−qo/θ′)
= 2−2N (1 + ww)−14
∫dc0
∏
e>0
(i−1dxedye
)e−ξ0c0+ξ0wye+
2θ′ poSxe+
2θ′ qoRyeΨ(c0, xn, yn) . (2.94)
By this Fourier transformation with respect to zero mode, the Grassmann parity of A(ξ0, xo, po, yo, qo)
and the corresponding |Ψ〉 coincide. The Moyal ⋆ product which is modified after Eqs.(2.92)(2.94)
is
⋆ := exp
(1
2
←−∂
∂ξΣ
−→∂
∂ξ
)= exp
(1
2
( ←−∂
∂ξ1σ′−→∂
∂ξ1+
←−∂
∂ξ2σ′−→∂
∂ξ2
)), (2.95)
ξ =
(ξ1
ξ2
), ξ1 =
(xo
po
), ξ2 =
(yo
qo
), Σ =
(σ′ 0
0 σ′
), σ′ = θ′
(0 1
1 0
).(2.96)
We define the trace in MSFT as integration over the “phase space”:
Tr A(ξ0, ξ) = detσ′∫dξ A(ξ0, ξ) = (−1)Nθ′2N
∫ ∏
o>0
(dxodpodyodqo) A(ξ0, ξ) . (2.97)
The Moyal field A(ξ0, ξ) which is mapped from |Ψ〉 by (2.94) is normalized as
∫dξ0Tr
((A(ξ0, ξ)
)†⋆
(∂
∂ξ0− θ′
2v∂
∂qo
)A(ξ0, ξ)
)= 〈Ψ|c0|Ψ〉 , (2.98)
20
where ∂∂ξ0− θ′
2 v∂∂qo
corresponds to −c0 (2.107).
It is convenient to introduce the bra 〈ξ0, ξ| as a state in Fock space such that the Moyal field is
related directly to the Fock space field via A(ξ0, ξ) = 〈ξ0, ξ|Ψ〉. This can be obtained from the bra
state 〈c0, xn, yn| (2.77) by Fourier transformation (2.94),
〈ξ0, ξ| = 〈ξ0, ξ1, ξ2| = 〈ξ0, xo, po, yo, qo|
= 2−2N (1 + ww)−14
∫dc0
∏
e>0
(i−1dxedye
)e−ξ0c0+ξ0wye+
2θ′ poSxe+
2θ′ qoRye〈c0, xn, yn|
= −2−2N (1 + ww)−14 〈Ω|c−1e
−ξ0(c0−√2wce)ecebe−cobo−2iξ1M
(o)0 ξ2−ξ1λ1−ξ2λ2 (2.99)
where we used notation:
M(o)0 =
(12 0
0 2θ′2SR
), λ1 =
(−i√2co
−2√2i
θ′ Sbe +2iθ′ Swξ0
), λ2 =
(−√2bo
−2√2
θ′ Rce
). (2.100)
This is the result given in the summary at the beginning of this section.
Examples Here we give some examples of string fields in MSFT.
For the conventional ghost number 1 vacuum c1|Ω〉 and SL(2, R) invariant vacuum |Ω〉, thecorresponding fields are given by
A0(ξ0, ξ) = 〈ξ0, ξ|c1|Ω〉 = 2−2N (1 + ww)−14 ξ0 e
−ixoyo−i 4
θ′2po(SR)
oo′qo′ , (2.101)
AΩ(ξ0, ξ) = 〈ξ0, ξ|Ω〉 = −i 2−2N+ 12 (1 + ww)−
14 ξ0 x1 e
−ixoyo−i4po(SR)oo′qo′ . (2.102)
We note that in the open string limit κe = e, κo = o, N = ∞, these expressions become singular.
For example, the coefficient of poqo′ is divergent,
(SR)oo′ =16 io+o′+2
π2o′
∞∑
e=2
e3
(e2 − (o′)2)(e2 − o2) = ±∞ . (2.103)
Therefore, it is advisable not to take the open string limit at the level of the state, but wait for
the end of a computation. We note that the physical quantities such as the scattering amplitude
become regular in this limit [10].
For the identity-like state in the Siegel gauge: |I〉 = NI e∑∞
n=1(−1)n c−n b−n c1|Ω〉, which is a delta
function with respect to the even modes in position basis, ΨI(c0, xn, yn) = NI
∏e>0(−4iδ(xe)δ(ye)),
the corresponding field is
AI(ξ0, xo, po, yo, qo) = (−1)N (1 + ww)−14NI ξ0 .
Except for the zero mode and the normalization factor, this AI is the identity element with respect
to the Moyal ⋆ product. The conventional identity state |I〉 (C.1) [2] (which is BRST invariant,
and not in the Siegel gauge) becomes more complicated in MSFT.
21
2.4 Oscillators
In this subsection, we obtain the Moyal images of the conventional oscillators which are used in
applications in MSFT. In this way we can write various operators in oscillator language and in
particular discuss the form of L0 and the butterfly state which came up in our work in [12].
2.4.1 Oscillators on the fields in MSFT
For an operator O which consists of bn, cn, acting on a state |Ψ〉 in Fock space, we define its Moyal
image βO, which is a differential operator acting on the Moyal field AΨ(ξ0, ξ) = 〈ξ0, ξ|Ψ〉, as follows
βOAΨ(ξ0, ξ) = 〈ξ0, ξ|O|Ψ〉 . (2.104)
For the basic operators, c0, b0, xo, yo, xe, ye, po, qo, pe, qe, this rule gives the corresponding operators
in MSFT
βc0 = − ∂
∂ξ0+θ′
2v∂
∂qo, βb0 = −ξ0 , (2.105)
βxo = xo , βyo = yo , βxe =θ′
2S∂
∂po, βye =
θ′
2T∂
∂qo, (2.106)
βpo =∂
∂xo, βqo =
∂
∂yo, βpe =
2
θ′Spo , βqe =
2
θ′R qo + weξ0 . (2.107)
The nonzero modes of the oscillators bn, cn become
βbe =1√2
∑
o>0
(2
θ′qoRo|e| − iǫ(e)
θ′
2S|e|o
∂
∂po
)+
1√2w|e|ξ0 =
∑
o
βboU−1−o,e +w′
eξ0 , (2.108)
βbo =1√2
(∂
∂y|o|− iǫ(o)x|o|
), (2.109)
βce =1√2
∑
o>0
(θ′
2T|e|o
∂
∂qo− iǫ(e) 2
θ′S|e|o po
)=∑
o
Ue,−oβco , (2.110)
βco =1√2
(y|o| − iǫ(o)
∂
∂x|o|
). (2.111)
In the first and third equations we introduced the symbols βbo,βco to denote the differential operators
βbo =1√2
(2
θ′q|o| − i
θ′
2ǫ(o)
∂
∂p|o|
), βco =
1√2
(θ′
2
∂
∂q|o|− iǫ(o) 2
θ′p|o|
). (2.112)
The even and the odd sets satisfy canonical anti-commutation relations
βbe, βce′ = δe+e′ , βbo, βco′ = δo+o′ , βbo, βco′ = δo+o′ . (2.113)
Using the above maps of operators, we can translate operators in the usual oscillator representation
into MSFT language. For example, the ghost number operator (we assigned ghost number 1 to
c1|Ω〉)Ngh =
∑
n≥1
(c−nbn − b−ncn) + c0b0 + 1 , (2.114)
22
is mapped to its MSFT image
Ngh =∑
o>0
(yo
∂
∂yo− xo
∂
∂xo+ po
∂
∂po− qo
∂
∂qo
)− ξ0
∂
∂ξ0+ 2 . (2.115)
2.4.2 Oscillator as a field
It is often useful to rewrite differential operators in Moyal space in terms of the star product. The
idea is to replace the derivative by the ⋆ -(super)commutator:8
∂
∂ξA = Σ−1[ξ, A⋆ . (2.116)
More concretely
∂
∂xoA =
1
θ′[po, A⋆ ,
∂
∂yoA =
1
θ′[qo, A⋆ ,
∂
∂poA =
1
θ′[xo, A⋆ ,
∂
∂qoA =
1
θ′[yo, A⋆ . (2.117)
This observation leads to the star product representation of the differential operators as follows
βboA =1√2
(βbo ⋆ A− (−1)|A|A ⋆ βb−o
), βboA =
1√2
(βbo ⋆ A+ (−1)|A|A ⋆ βb−o
), (2.118)
βcoA =1√2
(βco ⋆ A+ (−1)|A|A ⋆ βc−o
), βcoA =
1√2
(βco ⋆ A− (−1)|A|A ⋆ βc−o
), (2.119)
where we defined the fieldsβb,co in Moyal space that play the fundamental role of oscillators
βbo :=1
θ′q|o| −
i
2ǫ(o)x|o|, βco :=
1
2y|o| −
i
θ′ǫ(o)p|o| . (2.120)
The odd βb,co and even βb,ce differential operators are related to each other as in Eqs.(2.108,2.110).
Therefore we also define fields with even labels via the Bogoliubov transformation
βbe :=∑
o
βbo U−1−o,e , βce :=
∑
o
Ue,−oβco (2.121)
where the sum runs over odd integers from −2N + 1 to 2N − 1. These give the star product
representation of the even differential operators βb,ce
βbeA =1√2
(βbe ⋆ A+ (−1)|A|A ⋆ βb−e
)+ w′
eξ0A , (2.122)
βceA =1√2
(βce ⋆ A− (−1)|A|A ⋆ βc−e
). (2.123)
The fields βb,ce or βb,co satisfy the oscillator anticommutation relations with respect to the star
product
βbo, βco′⋆ = δo+o′ , βbe, βce′⋆ = δe+e′ . (2.124)
8We define the supercommutator as [A1, A2⋆ := A1 ⋆ A2 − (−1)|A1||A2|A2 ⋆ A1. Note that A←−∂∂ξ
= −(−1)|A|−→∂∂ξ
A .
23
But they do not anticommute with each other
βb−e, βco⋆ = U−1
o,−e , βbo, βc−e⋆ = U−e,o (2.125)
since they are related to each other by the Bogoliubov transformation given above.
One may regard the fields βb,c as the harmonic oscillators in Moyal space which act on the
string field A from either side by the star product. It is then natural to introduce the vacuum field
associated with the odd (and similarly the even) oscillators, as the field that satisfies the following
conditions under the star product
βbo ⋆ AB = βco ⋆ AB = AB ⋆ βb−o = AB ⋆ β
c−o = 0 , ∀o > 0 . (2.126)
By definition, this field is the Moyal image of the Fock space operator AB ∼ |0〉〈0|, where |0〉 is thevacuum state with respect to the oscillators βb,co . These are first order differential equations whose
solution is the gaussian
AB = ξ0 2−2N exp
(−∑
o>0
(ixoyo +
4i
θ′2poqo
)). (2.127)
If we write AB = ξ0AB, we see that AB is a projector (3.24) with respect to the Moyal ⋆ product:
AB ⋆ AB = AB . It turns out this is the butterfly projector that came up in other formulations of
string field theory [25, 26, 27] as shown in appendix E.
2.4.3 L0 and L0
In string field theory computations the zeroth Virasoro operator L0 plays a critical role since it
defines the propagator. In this section, we derive various forms of L0 in MSFT. In the usual
oscillator representation acting on Fock space, L0 is given by9
L0 =
∞∑
k=1
k(b−k ck + c−k bk) . (2.128)
In MSFT which is regularized by (N,κe, κo), we truncate the number of oscillators to 2N and
replace the frequencies by κe,o. Then the Moyal image of L0 becomes the following differential
operator
L0 =
2N∑
k=1
κk(βb−kβ
ck + βc−kβ
bk) (2.129)
=2N∑
k=1
κk + i∑
o>0
κo
(xoyo +
∂
∂xo
∂
∂yo+
4
θ′2poqo +
θ′2
4
∂
∂po
∂
∂qo
)
+4i
θ′2(1 + ww)
(∑
o>0
κovopo
)(∑
o′>0
vo′qo′
)+
2i
θ′(1 + ww)
(∑
o>0
voκopo
)ξ0 . (2.130)
9We take the convention such that L0(c1|Ω〉) = 0 fixes the constant that comes from normal ordering.
24
The last two terms come from the identities
SκeR = κoRR = κo + (1 + ww)κovv ,∑
e>0
Seoκewe = κoRwe = κov(1 + ww) . (2.131)
The similarity to the matter sector is enhanced by introducing an even basis(xbe, p
be, x
ce, p
ce
)which
is related to the odd basis through the following linear canonical transformation10
xbe := κ−1e Sxo , pbe := κeSpo , xce := Tyo , pce := Rqo . (2.132)
The Moyal ⋆ product (2.95) and trace (2.97) are invariant under this canonical transformation.
With the new variables, L0 is rewritten as
L0 =
2N∑
k=1
κk + i∑
e>0
(κ2e x
bex
ce +
∂
∂xbe
∂
∂xce+
4
θ′2pbep
ce +
θ′2
4κ2e
∂
∂pbe
∂
∂pce
)
− i
1 + ww
(∑
e>0
we∂
∂xbe
)(∑
e′>0
we′∂
∂xce′
)+
2i
θ′
(∑
e>0
wepbe
)ξ0 . (2.133)
Under this change of variables, the usual perturbative vacuum that was given in the odd basis
(2.101) becomes:
A0 = 2−2N (1 + ww)−14 ξ0 e
−ixbeRκoRxc
e−i 4
θ′2pbeκ−1e pce . (2.134)
Then the apparent divergence (2.103) of the coefficient at the limit κe = e, κo = o,N = ∞ does
not occur, since
|(RκoR)ee′ | =∣∣∣∣∣16 ie+e′ (ee′)2
π2
∞∑
o=1
1
o(e2 − o2)(e′2 − o2)
∣∣∣∣∣ <∞ . (2.135)
Following the ideas of the previous subsection, the differential operator L0 can be represented
in terms of star products by introducing the field L0 and the “midpoint correction” γ as follows
L0A = L0 ⋆ A+ A ⋆ L0 + γA , (2.136)
L0 = i∑
e>0
(κ2e2xbex
ce +
2
θ′2pbep
ce
)+
1
2
(∑
e>0
κe +∑
o>0
κo
)+
i
θ′
(∑
e>0
wepbe
)ξ0 (2.137)
=∑
e>0
κe
(βb−e ⋆ β
ce + βc−e ⋆ β
be
)− 1
2
(∑
e>0
κe −∑
o>0
κo
)+
i
θ′
(∑
e>0
wepbe
)ξ0 , (2.138)
γ = − i
1 + ww
(∑
e>0
we∂
∂xbe
)(∑
e′>0
we′∂
∂xce′
)(2.139)
where βb,ce can be rewritten in terms of the even mode variables in Eq.(2.132)
βbe =1
θ′pc|e| −
i
2ǫ(e)κex
b|e| , βce =
1
2xc|e| −
i
θ′ǫ(e)κ−1
e pb|e| . (2.140)
10We used these even modes in [12]. They are just a linear transformation of the odd modes in §2.3 and different
from the even modes which are considered in appendix D.
25
The ξ0-dependent term vanishes when L0 acts on the fields in the Siegel gauge. The field L0 is
multiplied with the star product, but γ is still a differential operator. It is possible to rewrite γ in
terms of star products (a double supercommutator)
γ A = − i
θ′2(1 + ww)
∑
e,e′>0
wewe′ [pbe, [p
ce′ , A⋆⋆ , (2.141)
but this does not have the single star product structure as the star products with L0 and therefore
γ cannot be absorbed into a redefinition of L0. As discussed in [12], γ depends only on a single
combination of modes in the direction of the vector we, which is closely related to the midpoint. If
it were not for the “midpoint correction” term γ, string field theory would reduce to a matrix-like
theory that would be exactly solvable, as shown in [12]. The above form of writing L0 focuses on
the γ term as the remaining difficult aspect of string field theory.
A similar structure exists in the canonically equivalent odd basis by using L′0, γ′
L0A = L′0 ⋆ A+ A ⋆ L′0 + γ′A ,
L′0 = i∑
o>0
κo
(1
2xoyo +
2
θ′2poqo
)+
1
2
(∑
e>0
κe +∑
o>0
κo
)+ (1 + ww)
i
θ′
(∑
o>0
voκopo
)ξ0
=∑
o>0
κo
(βb−o ⋆ β
co + βc−o ⋆ β
bo
)+
1
2
(∑
e>0
κe −∑
o>0
κo
)+ (1 + ww)
i
θ′
(∑
o>0
voκopo
)ξ0 ,
γ′ =4i
θ′2(1 + ww)
(∑
o>0
κovopo
)(∑
o′>0
vo′qo′
). (2.142)
3 Monoid and Neumann coefficients
The perturbative states and nonperturbative squeezed states in Fock space (perturbative vacuum,
sliver state, butterfly state and so on) are mapped to gaussian functions in Moyal space [9]. In
fermionic Moyal space, such as the basis ξ =(xbe, p
be, x
ce, p
ce
), the generic form of such a gaussian is
written as,
AN ,M,λ(ξ) = N e−ξMξ−ξλ , M = −M . (3.1)
As in the bosonic case, such shifted gaussians form a monoid algebra [9].11 A monoid is almost a
group except for the property of inverse. This implies that under star products the shifted gaussians
satisfy the properties of closure, identity and associativity. Although the generic gaussian has also
an inverse under star products, not all of them do (for example, projectors such as Eq.(2.127) do
not have an inverse). The identity element under star products is the natural number 1, which
corresponds to the trivial gaussian.
The monoid structure is an effective tool for computations in MSFT. In particular it was used
to calculate the product of n gaussian functions, whose trace gives the n-point vertex. A corollary
11Some issues on Moyal product are discussed in [20].
26
of this result is the determination of the Neumann coefficients for the n-string vertex. These
coefficients were computed in [2] from conformal field theory, but now they can be determined
by using only the Moyal product. The MSFT approach gives simple expressions for all Neumann
coefficients in terms of a single matrix teo = κ1/2e Teoκ
−1/2o . Neumann coefficients are not needed for
computations in MSFT, but the computation can be used to test the MSFT formalism. This was
used as successful test of MSFT in the matter sector [9].
In this section we will carry out a similar program in the ghost sector. While the treatment of
the unity elements becomes more subtle because of the zero mode, the closure of gaussian functions
(3.1) under the star product will be proved exactly the same way as in the bosonic sector. In
particular, the algebraic structure is formally the same. With this information, one can compute
the product of n fermionic gaussians and derive the Neumann coefficients by using this algebraic
machinery.
In particular, we verify consistency of the Moyal ⋆ product (2.95) with the conventional reflector
and the 3-string vertex in oscillator language. We will show the correspondence by including the
zero mode part,
〈Ψ1|Ψ2〉 ↔∫dξ0 Tr
(A1(ξ0, ξ) ⋆ A2(ξ0, ξ)
), (3.2)
〈Ψ1|Ψ2 ⋆W Ψ3〉 ↔
∫dξ
(3)0 dξ
(2)0 dξ
(1)0 Tr
(A1(ξ
(1)0 , ξ) ⋆ A2(ξ
(2)0 , ξ) ⋆ A3(ξ
(3)0 , ξ)
). (3.3)
3.1 Monoid structure for gaussian elements
We first derive the structure of the monoid (3.1) under the fermionic star product defined by
ξa, ξb = Σab for a general symmetric Σab. This matrix takes a block diagonal form Σ = diag (σ′, σ′)
in the bases we discussed so far, but it is useful to develop the formalism for any Σ. For the
moment, we suppress the ξ0-dependence because it is not relevant to the Moyal ⋆-product (2.95).
The structure of the monoid is summarized in the following algebra,
AN1,M1,λ1 ⋆ AN2,M2,λ2 = AN12,M12,λ12 , (3.4)
mi :=MiΣ , m = −ΣmΣ−1 , (3.5)
m12 =M12Σ = (1 +m2)m1(1 +m2m1)−1 + (1−m1)m2(1 +m1m2)
−1 , (3.6)
λ12 = (1−m1)(1 +m2m1)−1λ2 + (1 +m2)(1 +m1m2)
−1λ1 , (3.7)
N12 = N1N2 det12 (1 +m2m1) e
− 14
∑2a,b=1 λaΣKabλb , (3.8)
Kab =
((m−1
2 +m1)−1 (1 +m2m1)
−1
−(1 +m1m2)−1 (m2 +m−1
1 )−1
). (3.9)
To prove this formula, it is convenient to use Fourier transformation. We define
A(η) :=
∫dξ eξηA(ξ) , A(ξ) =
∫dη e−ξηA(η) . (3.10)
27
The ⋆ product is rewritten in terms of Fourier coefficients,
A1 ⋆ A2(ξ) =
∫dη1dη2e
− 12η1Ση2−ξη1−ξη2A1(η1)A2(η2) . (3.11)
For the gaussian AN ,M,λ (3.1), the Fourier transform is also gaussian:
˜AN ,M,λ(η) = N (det(2M))12 e−
14λM−1λe−
14ηM−1η+ 1
2ηM−1λ . (3.12)
The main result (3.4–3.9) follows by carrying out the gaussian integration over fermionic variables.
We note that Eq.(3.1) have exactly the same form as the bosonic case (Eqs.(3.11–3.17) in [9])
if we put d = −2. Similarly, the formula for the trace also related to the bosonic case as if d = −2
Tr(A(ξ)) = N (det(2m))12 e−
14λM−1λ . (3.13)
With this observation, we find that all the algebraic manipulations in [9] which use the structure
of the star product for monoids also apply in the ghost sector with no other modification.
In particular the product of n monoids with the sameM, but different λi,Ni is one of the useful
results that is used to compute the n-point vertex. The result in [9] is now adopted to the ghost
sector as follows
AN12···n,M (n),λ12···n := AN1,M,λ1 ⋆ AN2,M,λ2 ⋆ · · · ⋆ ANn,M,λn , (3.14)
m(n) =M (n)Σ =J−n
J+n, J±
n :=(1 +m)n ± (1−m)n
2, (3.15)
λ12···n =1
J+n
n∑
r=1
(1−m)r−1(1 +m)n−rλr , (3.16)
N12···n = N1N2 · · · Nn(det J+n )
12 exp
(−1
4Kn(λ)
), (3.17)
Kn(λ) =
n∑
r=1
λrΣJ−n−1
J+nλr + 2
n∑
r<s
λrΣ(1−m)s−r−1(1 +m)n+r−s−1
J+n
λs , (3.18)
Tr(AN12···n,M (n),λ12···n) = N1N2 · · · Nndet12 (2J−
n ) exp
−1
4
n∑
r,s=1
λrΣO(n)(s−r)(m)λs
, (3.19)
O(n)(r) (m) := O(n)
r mod n , (3.20)
O(n)0 (m) =
J+n−1
J−n
=(1 +m)n−1 + (1−m)n−1
(1 +m)n − (1−m)n, (3.21)
O(n)i (m) =
(1 +m)n−i−1(1−m)i−1
J−n
=2(1 +m)n−i−1(1−m)i−1
(1 +m)n − (1−m)n, (1 ≤ i ≤ n− 1) . (3.22)
This will be used in the next section to compute the Neumann coefficients as a by-product.
The algebraic structure is also used to construct projectors (such as sliver state or butterfly).
As in [9], the generic form of the projector in the ghost sector can be written as a particular class
of gaussian functions
AD,λ(ξ) = 2−2N e14λΣDΣλe−ξDξ−ξλ , (DΣ)2 = 1 , (3.23)
28
AD,λ ⋆ AD,λ(ξ) = AD,λ(ξ) , Tr(AD,λ(ξ)) = 1 . (3.24)
3.2 Neumann coefficients
In this section, we construct the Neumann coefficients by using the MSFT formalism for the n-
point vertices given above. The basic idea is to use the correspondence of vertices in the operator
formalism and in MSFT given by [9],
1〈Ψ1| ⊗ · · · ⊗ n〈Ψn|Vn〉 ∼ Tr(A1 ⋆ · · · ⋆ An) (3.25)
where Ar(ξ0, ξ) = 〈ξ0, ξ|Ψr〉 and |Ψr〉 = 〈Ψr|V2〉. For the ghost sector, we have to be careful in the
treatment of the zero mode in (3.25). We use this identification to express Neumann coefficients
by taking the following steps.
1. In Fock space we choose n coherent states for 〈Ψr|, r = 1, 2, · · · n, labelled by parameters
µ∗(r). The left hand side of (3.25) can be computed in the operator formalism. The result
takes the form of an exponential that contains a quadratic form in the parameters of the
coherent states µ(r). The Neumann coefficients that define |Vn〉 appear as the coefficients in
the quadratic form. We treat the Neumann coefficients as unknown matrices.
2. We calculate Ar(ξ0, ξ) = 〈ξ0, ξ|Ψr〉 which gives the Moyal image of the coherent states in the
form of monoids, with the λ(r) related to the parameters µ∗(r) of the coherent state.
3. We compute the right hand side of (3.25) by using the result in Eqs.(3.14–3.22). As in item
(1) this is also an exponential containing a quadratic form in the coherent state parameters
µ∗(r), but with the coefficients determined by the monoid algebra given above.
4. We compare the coefficients of the parameters in both sides and thus determine the Neu-
mann coefficients completely. They turn out to be simple functions of a single matrix
teo = κ1/2e Teoκ
−1/2o .
Throughout this subsection, we use the regularized framework (N,κe, κo) to make the algebraic
manipulation consistent. This gives a new generalization of Neumann coefficients since the new
expression includes arbitrary spectral parameters and arbitrary N . To compare to the Neumann
coefficients computed through conformal field theory, the open string limit (N =∞, κe = e, κo = o)
is taken at the end. Through analytic and numerical methods it is shown that these very different
looking forms of Neumann coefficients are indeed the same. This successful test of MSFT provides
confidence about its correctness and shows that MSFT is an alternative tool for computation in
string theory.
Coherent states Coherent states 〈Ψ| are defined by
〈Ψ|b† = 〈Ψ|µ∗b , 〈Ψ|c† = 〈Ψ|µ∗c , 〈Ψ|b0 = 〈Ψ|µ∗0 , (3.26)
29
They have the following explicit form
〈Ψ| = 〈Ω|c−1eµ∗b c+µ∗c b+µ∗0 c0 . (3.27)
The inner product between the standard n-string vertices |Vn〉 (appendix C, [2]) and the coherent
states 〈Ψr| is given as follows for n = 1, 2, 3
n = 1 : 〈Ψ|I〉 = π
2√2voκoµ
∗bo
(µ∗0 −
√2wµ∗be
)eµ∗cCµ∗b , (3.28)
n = 2 : 1〈Ψ1|2〈Ψ2|V2〉12 = (µ∗(1)0 − µ∗(2)0 )eµ
∗(1)c Cµ
∗(2)b
+µ∗(2)c Cµ
∗(1)b
= (µ∗(1)0 − µ∗(2)0 )e
12µ∗(1)εCµ∗(2)+ 1
2µ∗(2)εCµ∗(1) , (3.29)
n = 3 : 1〈Ψ1|2〈Ψ2|3〈Ψ3|V3〉123 = exp(−µ∗(r)c Xrsµ
∗(s)b − µ∗(r)c Xrs
0µ∗(s)0
)
= exp
(−1
2µ∗(r)εX rsµ∗(s) − µ∗(r)c Xrs
0µ∗(s)0
), (3.30)
where
µ∗ =
(µ∗cµ∗b
), ε =
(0 1
−1 0
), X rs =
(Xsr 0
0 Xrs
), C =
(C 0
0 C
), Cnm = (−1)nδn,m .
(3.31)
As we noted, the Neumann coefficients appear as the coefficients of quadratic functions of µ∗(r) in
the exponential.
Moyal image of coherent state We define a Moyal field which corresponds to the coherent
state 〈Ψ|. First, we have the corresponding ket |Ψ〉 by using the reflector (C.4):
|Ψ〉1 := 2〈Ψ|V2〉12 = ec(1)0 µ∗0+c†(1)Cµ∗b+µ∗cCb†(1) c
(1)1 |Ω〉1 . (3.32)
Then we get the Moyal field by using the ket 〈ξ0, ξ| (2.99) which defines the Moyal basis:
A(ξ0, ξ) := 〈ξ0, ξ|Ψ〉 = −2−2N (1 + ww)−14 (ξ0e
−√2µ∗cwµ∗0 + µ∗0)e
− 12µ∗εCµ∗−ξM0ξ−ξλ (3.33)
where we denoted
M0 =
(0 iM
(o)0
−iM (o)0 0
), λ =
−i√2µ∗co
−2√2i
θ′ Sµ∗be +2iθ′ Swξ0√
2µ∗bo2√2
θ′ Rµ∗ce
= 2K∗(µ∗ +Wξ0) , (3.34)
K∗ =
0 − i√2
0 0
0 0 −√2iθ′ S 0
0 0 0 1√2√
2θ′ R 0 0 0
, W =
0
0
− 1√2w
0
. (3.35)
It takes the form of the standard element of monoid although with the pre-factor N and the λ in
the exponent depend on the zero mode ξ0. We can apply the Moyal ⋆ product formula for monoids
which was developed in the previous subsection.
30
Explicit form of n-th product After using the results of (3.14–3.22), we have obtained the
trace formula for n-th product of Moyal fields which correspond to coherent states
Tr(A1(ξ(1)0 , ξ) ⋆ A2(ξ
(2)0 , ξ) ⋆ · · · ⋆ An(ξ
(n)0 , ξ))
= (−1)n2−2nN (1 + ww)−n4 det
12 (2J−
n )
n∏
r=1
(ξ(r)0 e−
√2µ∗(r)c wµ
∗(r)0 + µ
∗(r)0 )
× e− 12
∑nr=1 µ
∗(r)εCµ∗(r)− 14
∑nr,s=1 λ
(r)ΣO(s−r)(m0)λ(s)
= (−1)n2−2nN (1 + ww)−n4 det
12 (2J−
n )∏
r
(ξ(r)0 e−
√2µ∗(r)c wµ
∗(r)0 + µ
∗(r)0 )
× exp
1
2
n∑
r,s=1
µ∗(r)εC(K∗−12m0O(s−r)(m0)K∗ − δr,s)µ∗(s)
× exp
n∑
r,s=1
µ∗(r)εCK∗−12m0O(s−r)(m0)K∗Wξ
(s)0
, (3.36)
where we used the relations12
K∗Σ = −εCK∗−1m0 , m0 :=M0Σ , (3.37)
K∗−1m0K∗ =
(¯m∗−10 0
0 m∗−10
), m∗
0 :=√κm∗
0
1√κ=
(0 −S−T 0
), (3.38)
WεCK∗−12m0O(s−r)(m0)K∗W = 0 . (3.39)
Here m∗0 which defines m∗
0 has appeared in Eq.(A.62) in the context of the matter sector in MSFT.
We should emphasize that although we are using a notation similar to the one in the matter
sector, the meaning of m0 here is not the same as the one in the matter sector which is defined
by Eq.(A.60). However, because there are some relationships between the m0 in the matter and
ghost sectors there are some relations among Neumann coefficients in these sectors. Note that in
Eq.(3.36) we assigned different ξ0’s for each Moyal field in the trace. This prescription is necessary
to find agreement with Witten’s star product as we will see soon. Noting
ξ(r)0 e−
√2µ∗(r)c wµ
∗(r)0 + µ
∗(r)0 = δ(ξ
(r)0 + µ
∗(r)0 ) e−
√2µ∗(r)c wµ
∗(r)0 = δ(ξ
(r)0 + µ
∗(r)0 ) e2µ
∗(r)εCWµ∗(r)0 , (3.40)
we perform the ξ(r)0 -integrations for all zero modes, and find∫dξ
(n)0 dξ
(n−1)0 · · · dξ(1)0 Tr(A1(ξ
(1)0 , ξ) ⋆ A2(ξ
(2)0 , ξ) ⋆ · · · ⋆ An(ξ
(n)0 , ξ))
= (−1)n2−2nN (1 + ww)−n4 det
12 (2J−
n )
× exp
1
2
n∑
r,s=1
µ∗(r)εC(K∗−12m0O(s−r)(m0)K∗ − δr,s)µ∗(s)
× exp
n∑
r,s=1
µ∗(r)εC(2δr,s −K∗−12m0O(s−r)(m0)K∗)Wµ
∗(s)0
. (3.41)
12Similar relations were used to obtain the Neumann coefficients in matter sector. (§A.2.4 [9])
31
Comparison of coefficients Now, we consider the above formulas in the cases n = 1, 2, 3.
n = 1 case: Eq.(3.41) becomes
∫dξ0Tr A(ξ0, ξ) = −(1 + ww)
14 eµ
∗cCµ∗b . (3.42)
In this case from Eq.(3.28) there is a correspondence
∫dξ0 Tr A(ξ0, ξ) ∼ 〈Ψ|I〉 . (3.43)
up to pre-factor which comes from the b-ghost insertion in conventional operator formalism.[2] On
the other hand, up to a constant factor, we have
∫dξ0δ(ξ0)Tr A(ξ0, ξ) ∼ 〈Ψ|I〉 = µ∗0 e
µ∗cCµ∗b , (3.44)
for the identify-like state |I〉 which corresponds to the identity element for the reduced product
[21, 22]. In MSFT, the Moyal field 〈ξ0, ξ|I〉 is the identity element in the Siegel gauge. In this
sense, |I〉 appears naturally rather than the BRST-invariant |I〉 in the context of MSFT.
n = 2 case: Using Eq.(3.36), we get
∫dξ
(2)0 dξ
(1)0 δ(ξ
(1)0 − ξ
(2)0 )Tr (A1(ξ
(1)0 , ξ) ⋆ A2(ξ
(2)0 , ξ))
= (µ∗(1)0 − µ∗(2)0 ) e
12µ∗(1)εCµ∗(2)+ 1
2µ∗(2)εCµ∗(1) . (3.45)
From Eq.(3.29) we have obtained the correspondence
∫dξ
(2)0 dξ
(1)0 δ(ξ
(1)0 − ξ
(2)0 )Tr(A1(ξ
(1)0 , ξ) ⋆ A2(ξ
(2)0 , ξ)) = 1〈Ψ1|2〈Ψ2|V2〉12 . (3.46)
In this case the normalization also coincides. We can interpret∫dξ
(2)0 dξ
(1)0 δ(ξ
(1)0 − ξ
(2)0 ) as the
pre-factor (c(1)0 + c
(2)0 ) in the form of 1〈c(1)0 , x
(1)n , y
(1)n |2〈c(2)0 , x
(2)n , y
(2)n |V2〉12 (C.5). Eq.(3.46) can be
rewritten as∫dξ0Tr(A1(ξ0, ξ) ⋆ A2(ξ0, ξ)) = 〈Ψ1|Ψ2〉 , (3.47)
for |Ψ2〉1 := 2〈Ψ2|V2〉12. This is consistent with the normalization (2.98) which we adopted to fix
the map from the conventional field to the Moyal field (2.94).
n = 3 case: We can identify the Neumann coefficients for the nonzero modes by comparing
Tr(A1(ξ(1)0 , ξ) ⋆ A2(ξ
(2)0 , ξ) ⋆ A3(ξ
(3)0 , ξ)) ∼ 〈Ψ1|〈Ψ2|〈Ψ3|V3〉 . (3.48)
From Eqs.(3.36,3.30) we get the Neumann coefficients in MSFT:
X rs = −C(K∗−12m0O(s−r)(m0)K∗ − δr,s).
32
More explicitly
X(0) = −C m∗−20 − 1
m∗−20 + 3
, X(+) = −C 2(1 + m∗−10 )
m∗−20 + 3
, X(−) = −C 2(1 − m∗−10 )
m∗−20 + 3
. (3.49)
To identify the Neumann coefficients including the zero mode part, we should perform the ξ0
integration:∫dξ
(3)0 dξ
(2)0 dξ
(1)0 . We can interpret that this comes from the pre-factor:
(c(1)0 − wy(1)e )(c
(2)0 − wy(2)e )(c
(3)0 − wy(3)e ) = c(1)c(2)c(3)
in the form
1〈c(1)0 , x(1)n , y(1)n |2〈c(2)0 , x(2)n , y(2)n |1〈c(3)0 , x(3)n , y(3)n |V2〉123(C.7). In fact, by identifying13 Eq.(3.41) with Eq.(3.30):
∫dξ
(3)0 dξ
(2)0 dξ
(1)0 Tr(A1(ξ
(1)0 , ξ) ⋆ A2(ξ
(2)0 , ξ) ⋆ A3(ξ
(3)0 , ξ)) ∼ 〈Ψ1|〈Ψ2|〈Ψ3|V3〉 (3.50)
up to constant factor, we obtain14
X(0)0 = −2εCW + εCK∗−1 2 + 2m2
0
m20 + 3
K∗W =4
m∗−20 + 3
w√2,
X(+)0 = εCK∗−1 2 + 2m0
m20 + 3
K∗W = −2− 2m∗−10
m∗−20 + 3
w√2, (3.51)
X(−)0 = εCK∗−1 2− 2m0
m20 + 3
K∗W = −2 + 2m∗−10
m∗−20 + 3
w√2.
The Neumann coefficients X(0,±),X(0,±)
0 agree with Eq.(A.68) which was obtained by using the
trace of 6 coherent states in the matter sector. This implies that the Gross-Jevicki nonlinear
relations for Neumann coefficients in MSFT are all satisfied for arbitrary (N,κe, κo), as was shown
in [9]. Namely, our Moyal star product is consistent with the conventional Witten star product
in both the matter and ghost sectors.15 We have confirmed the correspondence between Moyal ⋆
product in MSFT and Witten’s one (⋆W ) :
∫dξ
(3)0 dξ
(2)0 dξ
(1)0 Tr(A1(ξ
(1)0 , ξ) ⋆ A2(ξ
(2)0 , ξ) ⋆ A3(ξ
(3)0 , ξ)) ↔ 〈Ψ1|Ψ2 ⋆
W Ψ3〉 (3.52)
up to constant factor for κe = e, κo = o,N = ∞. As we will show in the next subsection we have
numerical confirmation that the generalized Neumann coefficients in MSFT for arbitrary κe, κo, N
converge to the conventional one in the operator formalism when we take the limit.
13The ghost zeromode ξ0 dependence is similar to momentum p0 dependence in matter sector. But in this case, we
do not need “momentum conservation factor” δ(ξ(1)0 +ξ
(2)0 +ξ
(3)0 ). This fact correspond to the lack of δ(b
(1)0 +b
(2)0 +b
(3)0 )
factor in Ref.[24] Eq.(2.18) which gives the correct 3-string vertex in oscillator representation.
14We used the notation: w =
(we
0
).
15This also implies that Moyal ⋆ product (2.95) is essentially the same as the reduced product in [16, 21, 22] which
was defined by omitting ghost zero mode-dependence in original Witten’s star product.
33
3.3 Numerical comparison of Neumann coefficients
In this subsection we compare the generalized Neumann coefficients derived algebraically in the
Moyal star formalism for any κe, κo, N, with the independent computation from the point of
view of conformal field theory [2] valid at κe = e, κo = o,N = ∞. We summarize the Neumann
coefficients computed from CFT in appendix G.1, together with some differences in the convention.
In [9] we have already given an analytic proof that our algebraic expression of Neumann coeffi-
cients coincides with the exact value in [2] in the limit, by comparing the spectroscopy of Neumann
coefficients. Namely, in our case by diagonalizing the matrix teo = κ1/2e Teoκ
−1/2o we diagonalize the
Neumann coefficients for n-point vertices, since they all depend on the same matrix t. The eigen-
values obtained in this way for the case of the 3-point vertex in the limit κe = e, κo = o,N = ∞coincide with the corresponding eigenvalues obtained from Neumann spectroscopy in [13].
A numerical study provides another approach to confirm that the Moyal star and CFT calcula-
tions agree in the limit. In the following numerical analysis we show that there is agreement in the
limit, and furthermore that there is a clear universal behavior of the approach to the limit as a func-
tion of N . In the tables given in appendix G.2 we give the MSFT results for the numerical values
of the Neumann coefficient M(0)ee′ (N) in the matter sector, and the Neumann coefficient X
(0)ee′ (N)
in the ghost sector for e, e′ = 2, 4, 6, 8, at different values of the cut-off parameter N . We set the
spectral parameters as κe = e, κo = o. The expression of the Neumann coefficients in the Moyal star
computation is given in (5.32–5.34) in [9] for the matter sector, and Eqs.(3.49,3.51) in this paper
for the ghost sector. In the tables we write the ratio with their limiting value,M(0)ee′(N)/M(0)
ee′ (cft)
and X(0)ee′ (N)/X
(0)ee′ (cft), where the limiting value is taken as the CFT value given in [2]. The tables
at different values of N clearly show the convergence
limN→∞
M(0)ee′ (N)
M(0)ee′ (cft)
= 1 , limN→∞
X(0)ee′ (N)
X(0)ee′ (cft)
= 1 . (3.53)
Namely in the open string limit, the Neumann coefficients derived algebraically in MSFT becomes
identical with their analytic value computed in CFT.
We note that the convergence of the Neumann coefficients of the ghost sector is much slower
than those of matter sector. However log-log plot of |M(N)/M(cft) − 1| against N clearly shows
that the deviation scales as power of N with a very good accuracy.
As examples, we write the fitting of (2, 2) and (2, 4) components of above ratios as16
M(0)22 (N)
M(0)22 (cft)
∼ 1 + 1.33 ·N−1.34 ,M(0)
24 (N)
M(0)24 (cft)
∼ 1 + 2.38 ·N−1.36 , (3.54)
X(0)22 (N)
X(0)22 (cft)
∼ 1 + 0.834 ·N−0.669 ,X
(0)24 (N)
X(0)24 (cft)
∼ 1 + 1.22 ·N−0.684 . (3.55)
16These are based on the numerical data for N = 20, 50, 100, 200, 400.
34
We have numerically checked that all the Neumann coefficients including the zero mode behave
exactly the same way as above,
M(0,±)(N)
M(0,±)(cft)− 1 ∼ α(0,±)m
nm N−βm ,X(0,±)(N)
X(0,±)(cft)− 1 ∼ α(0,±)gh
nm N−βgh, (3.56)
where the coefficients α’s are order one quantity which depends on the type of the Neumann
coefficients. On the other hand, the power βm and βgh are universal for matter and ghost sector.
In the numerical study so far, βm ∼ 1.33 and βgh ∼ 0.67 for all types of Neumann coefficients.
We suspect that there may be an analytic evaluation of the deviations which will prove such a
systematic behavior. In any case, Eq.(3.56) gives a useful numerical estimate of the deviation at
finite N from the N =∞ values.
4 Applications
In this section, we consider the applications of the Moyal star formulation in the ghost sector. We
discuss two topics which are essential in the development of MSFT.
The first issue is the derivation of the regularized string field theory action in the Siegel gauge
including ghosts
S = −∫ddxTr
(1
2α′ A ⋆ (L0 − 1)A+g
3A ⋆ A ⋆ A
). (4.1)
The regularized version was the starting point of our recent discussions in [10, 12].
The second issue is the derivation of the Feynman rules in the ghost sector. Together with our
previous work on Feynman diagrams in the matter sector [10] this provides the complete set of
Feynman rules. We show some explicit examples of computations of amplitudes.
4.1 Regularized MSFT action and equation of motion
We start from Witten’s string field theory action in the operator formulation
S =1
2〈Ψ|QB|Ψ〉+
1
3〈Ψ|Ψ ⋆W Ψ〉 . (4.2)
QB is BRST operator, which may be written by separating out the b0, c0 zero modes
QB = c0(L0 − 1) + 2Xb0 + Q , (4.3)
with
L0 = Lmatter0 + Lghost
0 :=1
2α20 +
∞∑
n=1
α−nαn +∞∑
n=1
n(b−ncn + c−nbn) , (4.4)
X = −∞∑
n=1
nc−ncn , (4.5)
35
Q =∑
n 6=0
c−nLmattern +
∑
m,n,m+n 6=0
m− n2
cmcnb−m−n , (4.6)
Lmattern =
1
2
∞∑
m=−∞α−mαn+m . (4.7)
By imposing the Siegel gauge condition b0|Ψ〉 = 0 we obtain the gauge fixed action
S =1
2〈Ψ|c0(L0 − 1)|Ψ〉+ 1
3〈Ψ|Ψ ⋆W Ψ〉 . (4.8)
In the regularized version of MSFT with cut-off parameters (N,κe, κo), we cannot write a nilpotent
QB operator, at least technically for the time being, because the conformal symmetry is explicitly
broken when the parameters (N,κe, κo) are not at their limiting values. Of course, any other
approach that attempts to work with a finite number of modes (such as level truncation) suffers
from the same problem. For complete control, what seems to be desirable is the construction of a
finite dimensional Lie algebra that would be a substitute for the Virasoro algebra at finite N, and
which would tend to the Virasoro algebra at infinite N. If such an algebra could be constructed,
then a regulated version of QB at finite N would be straightforward, at least in MSFT.
On the other hand we have seen in numerous cases by now that the regulator is indispensable.
With this restriction, we are forced to work with the gauge fixed action Eq.(4.8) where the trunca-
tion of the oscillators can be made self-consistently. In the open string limit, we recover the original
gauge fixed action which is equivalent to the original gauge invariant action Eq.(4.2).
In the following we rewrite the action (4.8) in the Moyal language. We use a field A(x, ξ0, ξ) =
ξ0A(x, ξ) in the Siegel gauge which is related to a conventional string field Ψ by Fourier transfor-
mation (A.28)(2.94). The kinetic term is rewritten as,
〈Ψ|c0(L0 − 1)|Ψ〉 =
∫(−dξ0)
∫ddxTr
(A(x, ξ0, ξ) ⋆ βc0(L0 − 1)A(x, ξ0, ξ)
)
=
∫ddxTr (A(x, ξ) ⋆ (L0 − 1)A(x, ξ)) (4.9)
where17 L0 = Lmatter0 + Lghost
0 is given by
Lmatter0 =
1
2β20 −
d
2Tr κ− 1
4DξM
−10 κDξ + ξκM0ξ ,
Lghost0 = Tr κgh − 1
2
∂
∂ξb
(Mgh
0
)−1κgh
∂
∂ξc+ 2ξbκghMgh
0 ξc
= Tr κgh − 1
4
∂
∂ξghε(Mgh
0
)−1κgh
∂
∂ξgh+ ξghεκghMgh
0 ξgh , (4.10)
β0 = −ils∂
∂x, Dξ =
(∂
∂xe− iwe
lsβ0
∂∂pe
), κ =
(κe 0
0 TκoR
), M0 =
(κe
2l2s0
0 2l2sθ2Tκ−1
o T
),
17In this section, we use the variable ξgh = (ξbe, ξce) = (xb
e, pbe, x
ce, p
ce) which was introduced in Eq.(2.132) because
this makes Lghost0 most similar to the Lmatter
0 .
36
κgh =
(RκoT 0
0 κe
), ε =
(0 1
−1 0
), Mgh
0 =
(i2RκoR 0
0 2iθ′2κ−1e
)
and the Moyal ⋆ product and the trace are
⋆ = exp
(1
2
←−∂
∂ξσ
−→∂
∂ξ+
1
2
←−∂
∂ξghΣ
−→∂
∂ξgh
), Tr =
detσ′
|det(2πσ)|d/2∫d2Ndξ d4N ξgh , (4.11)
σ = iθ
(0 1
−1 0
), Σ =
(σ′ 0
0 σ′
), σ′ = θ′
(0 1
1 0
).
On the other hand, the cubic term of the action becomes
〈Ψ|Ψ ⋆W Ψ〉 = µ−13
∫dξ
(3)0 dξ
(2)0 dξ
(1)0
∫ddxTr
(A(x, ξ
(1)0 , ξ
)⋆ A
(x, ξ
(2)0 , ξ
)⋆ A
(x, ξ
(3)0 , ξ
))
= µ−13
∫ddxTr (A (x, ξ) ⋆ A (x, ξ) ⋆ A (x, ξ)) (4.12)
where
µ3 = −22N(d−2)(1 + ww)−d8+ 3
4 (det(3 + tt))−d(det(1 + 3tt))2 , t := κ1/2e Tκ−1/2o . (4.13)
After an appropriate rescaling of A, we obtain the gauge fixed action (4.1) in MSFT language:
S = −∫ddxTr
(1
2α′
∫(−dξ0)A(x, ξ0, ξ) ⋆ βc0(L0 − 1)A(x, ξ0, ξ)
+g
3
∫dξ
(3)0 dξ
(2)0 dξ
(1)0 A
(x, ξ
(1)0 , ξ
)⋆ A
(x, ξ
(2)0 , ξ
)⋆ A
(x, ξ
(3)0 , ξ
))
= −∫ddxTr
(1
2α′ A(x, ξ) ⋆ (L0 − 1)A(x, ξ) +g
3A(x, ξ) ⋆ A(x, ξ) ⋆ A(x, ξ)
)(4.14)
where A (x, ξ0, ξ) = ξ0A(x, ξ) is a Grassmann odd field in the Siegel gauge. The conventional reality
condition of the string field in Fock space 〈V2|Ψ〉 = (|Ψ〉)† is simply given by the usual reality of
the field in Moyal space A (x, ξ)† = A (x, ξ).
The equation of motion in MSFT becomes
(L0 − 1)A(x, ξ) + α′g A(x, ξ) ⋆ A(x, ξ) = 0, (4.15)
which corresponds to the equation of motion in the Siegel gauge (L0 − 1)Ψ + b0Ψ ⋆W Ψ = 0 in
conventional language. The counterpart of the usual classical equation of motion QBΨ+Ψ⋆WΨ = 0
is difficult to express in the cut-off theory as we already commented. Similarly we meet a similar
difficulty to express the BRST invariance condition QΨ + b0c0Ψ ⋆W Ψ = 0 in the Siegel gauge in
MSFT at this stage.
4.2 Computing Feynman graphs including fermionic ghost sector
We have defined the gauge fixed action Eq.(4.14) in MSFT language. Based on it, we discuss the
Feynman rules in MSFT and show simple examples explicitly. Computations in the matter sector
have already been presented in [10].
37
Vertex In MSFT the n-string interaction vertex is represented by n-th Moyal ⋆ product and its
trace (4.11). In Fourier basis, eiξηe−ξghηgh , this amounts to a phase factor to represent the vertex
as follows:
Tr((eiξη1e−ξghηgh1 ) ⋆ · · · ⋆ (eiξηne−ξghηghn )
)
=(−)N (θ′)2N
(2πθ)Ndexp
−1
2
∑
i<j
ηiσηj −1
2
∑
i<j
ηghi Σηghj
×(2π)2Ndδ2Nd(η1 + · · · + ηn) δ4N (ηgh1 + · · ·+ ηghn ) . (4.16)
The constant factor comes from |det(2πσ)|−d/2 det σ′ in the definition of the trace.
Propagator It is convenient to introduce the propagator ∆(η, η′, τ, p) in Fourier basis. This was
computed in the matter sector in [10]. Here we give the complete form, including the fermionic
ghost sector
∆(η, η′, τ, p) :=
∫d2Ndξ
(2π)2Ndd4N ξgh (e−iξηeξ
ghηgh) e−τL0(p)(eiξη′e−ξghη′gh)
= ed−22
τ∑
n>0 κn+12(1+ww)l2sp
2
∫d2Ndξ
(2π)2Ndd4Nξgh e−iξηeξ
ghηgh
× e−ilsτ2
pweηxe+τ4η′M−1
0 κη′+τ ∂∂η′ κM0
∂∂η′ eiξη
′
× eτ4η′ghεMgh−1
0 κghη′gh−τ ∂
∂η′ghεκghMgh
0∂
∂η′gh e−ξghη′gh . (4.17)
Here L0(p) is given by setting β0 = lsp in Eq.(4.10). Using Eqs.(F.5,F.8 ), we obtain the propagator
in the bc ghost sector
∆(η, η′, τ, p) = g(τ, p) e−ηF (τ)η−η′F (τ)η′+2ηG(τ)η′+(η+η′)H(τ,p)
× eηghF gh(τ)ηgh+η′ghF gh(τ)η′gh−2η′ghGgh(τ)ηgh (4.18)
where
g(τ, p) =
(θ
2π
)Nd (−1)Nθ′2N
(1 + ww)d+24
×(∏
e>0
(1− e−2τκe)∏
o>0
(1− e−2τκo)
)− d−22
e−(
τ2+w
tanh( τκe2 )
κew
)l2sp
2
,
F (τ) =1
4M−1
0 (tanh(τ κ))−1 =
(l2s2κe
(tanh(τκe))−1 0
0 θ2
8l2sRκo(tanh(τκo))
−1R
),
G(τ) =1
4M−1
0 (sinh(τ κ))−1 =
(l2s2κe
(sinh(τκe))−1 0
0 θ2
8l2sRκo(sinh(τκo))
−1R
),
H(τ, p) =tanh(τκe/2)
κewl2sp , (4.19)
38
F gh(τ) =1
4εMgh−1
0 (tanh τ κgh)−1 = ε
(− i
2Tκ−1o (tanh τκo)
−1T
− iθ′28 κe(tanh τκe)
−1
),
Ggh(τ) =1
4εMgh−1
0 (sinh τ κgh)−1 = ε
(− i
2Tκ−1o (sinh τκo)
−1T
− iθ′28 κe(sinh τκe)
−1
).
The ghost structure of the quadratic term in the exponent is similar to the matter one.
1-loop vacuum amplitude By taking the trace of the propagator Eq.(4.18), we have
∫ddpTr e−τ(L0−1) =
∫ddp
∫d2Ndη d4Nηgh eτ ∆(η, η, p, τ)
= (2π)d2 l−d
s τ−d2 eτ
∏
e>0
(1− e−τκe)−(d−2)∏
o>0
(1− e−τκo)−(d−2) . (4.20)
This reproduces the expected partition function, with the correct spectrum, including the ghost
contribution. If we take the open string limit κe = e, κo = o,N = ∞ naively in the formula of L0
(A.47,2.133), namely at the Lagrangian level, we lose the information on odd spectrum. This is one
of the indications that the γ term plays a nontrivial role. Of course, we obtain the correct limiting
partition function by taking the limit at the last stage of the computation, which is given above.
External state As external states in Feynman graphs it is enough to consider monoid elements
such as
AN ,M,λ,Mgh,λgh = N eipxe−ξMξ−ξλe−ξghMghξgh−ξghλgh
. (4.21)
In fact, we can compute various perturbative diagrams by preparing a particular class of gaussian
external states given by M = M0,Mgh = εMgh
0 , where these matrices were given in Eqs.(4.10). If
we also take λ = (−iwep, 0), λgh = 0, this external field represents perturbative vacuum c1|p,Ω〉
with momentum p, which represents the perturbative tachyon with proper normalization
Ap(ξ) = 2N(d−2)(1 + ww)−d+28 eipxe−ξM0ξ+ipwexee−ξghεMgh
0 ξgh , Tr(Ap(ξ)† ⋆ Ap(ξ)) = 1 .(4.22)
We omitted an overall ξ0 because it drops out in the Siegel gauge action (4.14). Excited states (that
correspond to polynomials multiplying thisAp(ξ)) can be obtained by differentiating Ap(ξ)e−ξλ−ξghλgh
with respect to general λ, λgh appropriately, and then setting λ = (−iwep, 0), λgh = 0. Therefore
an explicit computation of Feynman graphs with general λ, λgh,M,Mgh has many physical appli-
cations.
τ-evolved monoid element It is convenient to have a τ -evolved formula for a gaussian Eq.(4.21)
to compute Feynman graphs in ξ-basis [10]. We can derive it explicitly by evaluating:
e−τL0AN ,M,λ,Mgh,λgh(ξ)
39
=
∫d2Ndη d4Nηgh e−iξηeξ
ghηgh(∫
d2Ndη′ d4Nη′gh∆(η, η′, τ, p) AN ,M,λ,Mgh,λgh(η′)),
AN ,M,λ,Mgh,λgh(η) =
∫d2Ndξ
(2π)2Ndd4N ξgh e−iξηeξ
ghηghAN ,M,λ,Mgh,λgh(ξ) . (4.23)
By gaussian integration we have the following formula
e−τL0AN ,M,λ,Mgh,λgh(ξ) = NNm(τ)N gh(τ) eipxe−ξM(τ)ξ−ξλ(τ)e−ξghMgh(τ)ξgh−ξghλgh(τ) (4.24)
where
M(τ) =[sinh τ κ+
(sinh τ κ+M0M
−1 cosh τ κ)−1](cosh τ κ)−1M0 , (4.25)
λ(τ) =[(cosh τ κ+MM−1
0 sinh τ κ)−1
(λ+ iwp)]− iwp , (4.26)
Nm(τ) =e−
12l2sp
2τ exp[14
(λ+ ipw
)(M + coth τ κ M0)
−1 (λ+ iwp)]
det(12
(1 +MM−1
0
)+ 1
2
(1−MM−1
0
)e−2τκ
)d/2 (4.27)
for the matter sector [10]18 and
Mgh(τ) =
[sinh τ κgh +
(sinh τ κgh + εMgh
0 Mgh−1 cosh τ κgh)−1
](cosh τ κgh)−1εMgh
0 , (4.28)
λgh(τ) =[cosh τ κgh −MghεMgh−1
0 sinh τ κgh]−1
λgh , (4.29)
N gh(τ) = e−14λgh(Mgh+coth(τκgh)εMgh
0 )−1λgh
×[det
(1
2(1−MghεMgh−1
0 ) +1
2(1 +MghεMgh−1
0 )e−2τκgh
)] 12
(4.30)
for ghost sector. When we consider a class of monoid such that ξghMghξgh is SU(1, 1)-symmetric
and twist even19 then the evolved M(τ) also has this symmetry. We can see this explicitly by noting
that f(κ)Mgh0 is a block diagonal and symmetric matrix (where f(x) is an arbitrary function). In
this case with Mgh = εMgh′ , where Mgh′ is a 2N × 2N matrix, the above formula becomes
Mgh(τ) =ε
[sinh τ κgh +
(sinh τ κgh +Mgh
0 Mgh′−1 cosh τ κgh)−1
](cosh τ κgh)−1Mgh
0 , (4.31)
λgh(τ) =[cosh τ κgh +Mgh′Mgh−1
0 sinh τ κgh]−1
λgh , (4.32)
N gh(τ) = e−14λghε(Mgh′+coth(τκgh)Mgh
0 )−1λgh
× det
(1
2(1 +Mgh′Mgh−1
0 ) +1
2(1−Mgh′Mgh−1
0 )e−2τκgh
). (4.33)
We can use this reduced formula to compute the τ -evolved monoid of n-th product of the pertur-
bative vacuum: e−τL0
(A0(ξ)e
−ξλ1 ⋆ · · · ⋆ A0(ξ)e−ξλn
)because the coefficient matrix M
(n)0 (3.15)
in the quadratic term in the exponent is proportional to ε.
With the above preparation, we have all that is needed to compute the ghost contribution in
various amplitudes, by following the methods that were developed in the matter sector in [10].18We supposed that M is Lorentz symmetric.19In §H, we defined SU(1, 1)-symmetric and twist even monoid element which has λgh = 0 (H.9). Here we permit
λgh 6= 0 case.
40
4-tachyon amplitude The 4 point amplitude for tachyons is computed by putting together
several diagrams that are related to each other by permutations of the external legs. For a typical
4-pt diagram 21>−<3
4 the MSFT expression is
12A34 =
∫ddxTr
(e−τL0 (A1(ξ) ⋆ A2(ξ)) ⋆ (A3(ξ) ⋆ A4(ξ))
)(4.34)
where τ is the length of the propagator. When Ai(ξ), (i = 1, · · · , 4) are gaussians, we can compute
this quantity easily by taking the ⋆ product between the pairs of gaussians (3.4–3.9), evolving by
τ (4.31), and computing the trace (3.13). At each step we only use the properties of the monoid.
In the case of tachyons, the matter contribution was already computed in [10]. For the ghost
contribution, we set the external field to Aghi (ξ) = 2−2N (1 + ww)−
14 e−ξghεMgh
0 ξgh and obtain
12Agh34 = (det(2mgh
0 ))−1(det(1 − (mgh0 )2))2
×
det
4 sinh τ κgh
(cosh τ κgh +
2
1 + (mgh0 )2
sinh τ κgh
)2
− 1
−1
eτκgh
−1
= 2−8N (1 + ww)32 (det(1 + 3tt))4
× det
(1−
(tt− 1
1 + 3tte−κeτ
)2)det
(1−
(tt− 1
1 + 3tte−κoτ
)2)
(4.35)
where mgh0 =Mgh
0 σ′, t = κ1/2e Tκ
−1/2o . Including the matter sector [10] we obtain
12A34 = 24(d−2)N (1 + ww)−d4+ 3
2 (det(1 + 3tt))4 (det(3 + tt))−2d (2π)dδd(p1 + p2 + p3 + p4)
×det
(1−
(tt−11+3tt e
−κeτ)2)
det
(1−
(tt−11+3tt e
−κoτ)2)
[det
(1−
(tt−13+tt e
−κeτ)2)
det
(1−
(tt−13+tt e
−κoτ)2)] d
2
× exp
(−1
2l2s(p1 + p2)
2(τ + α(τ)) + l2s(p1 + p3)2β(τ) +
1
2l2s
4∑
i=1
p2i γ(τ)
)(4.36)
where
α (τ) = vκ− 1
2o
[t
(1 + tt+
1
2(1 + tt) coth
τκe2
(1 + tt)
)−1
t (4.37)
+
(1 + tt+
1
2(1 + tt) coth
τκo2
(1 + tt)
)−1]κ− 1
2o v ,
β (τ) = 2vκ− 1
2o
(4 sinh τκo + (1 + tt) sinh τκo (1 + tt) (4.38)
+ 2(1 + tt) cosh τκo + 2cosh τκo (1 + tt)
)−1
κ− 1
2o v ,
γ (τ) = −vκ−12
o cothτκo2
(2 + (1 + tt)coth
τκo2
)−1κ− 1
2o v . (4.39)
41
We note that the matrices tt, tt in the determinant factors in the matter sector come out inverted in
the ghost sector. By integrating with the measure∫∞o dτeτ and adding permutations of the diagram,
we should reproduce the Veneziano amplitude20 when all the tachyons are on-shell, l2sp2i = 2 in the
open string limit κe = e, κo = o,N =∞.
Our formula (4.36) has a counterpart in the operator formalism. Although our expressions are
simpler it is not easy to compare results analytically because of the different formalisms. We have
managed to compare and agree with the determinant factor available in the computations in [29]
in the operator formalism, by inserting the MSFT ghost Neumann coefficients given in (3.49) and
matter Neumann coefficients taken from [9]
M(0) =m∗2
0 − 1
m∗20 + 3
, CX(0) =√κm∗2
0 − 1
1 + 3m∗20
1√κ, m∗2
0 =
(tt 0
0 tt
). (4.40)
The difference of normalization compared to [29] comes from that of cubic term of the action (4.13).
We note that so far it has not been demonstrated yet that either the operator formalism or the
MSFT approach reproduce the Veneziano amplitude analytically, although this is expected to be
true.
5 Discussion
In this paper we provided the details of the Moyal star formulation for fermionic ghosts. Following
the similar construction in the matter sector, the split string formalism was used as an intermediate
step. However, as in the matter sector, the midpoint needed additional considerations to insure that
MSFT is in agreement with the operator formulation of string field theory. MSFT then provides an
alternative method of computation in string field theory which is in many ways simpler and more
efficient.
The regularization of the fermionic ghost sector, which is needed to avoid the associativity
anomaly, is made consistently with the matter sector. The correctness of the formulation, including
the regularization, was tested by computing the Neumann coefficients by using MSFT methods and
comparing them to an independent computation that relies on conformal field theory. The MSFT
result generalizes the Neumann coefficients by computing them for any set of oscillator frequencies
κe, κo for any finite number of oscillators 2N. These agree with conformal theory results in the open
string limit κe = e, κo = o, N = ∞. The agreement was established both analytically as well as
numerically.
In numerical study of string field theory one necessarily deals with a finite number of modes. One
may debate which version of Neumann coefficients is more consistent for such numerical study: the
finite N version of the Neumann coefficients given in MSFT, or the level truncation of the infinite
Neumann matrices practiced in previous literature? The numerical analysis that we provided could
20The expressions here are slightly further simplified than the ones in [10].
42
be helpful in understanding the issue and developing the most appropriate numerical approximation
scheme. We hope to address this point, together with numerical studies of certain quantities in the
near future.
The regularized MSFT formulation is now complete in the Siegel gauge. It has already been
applied to the computation of perturbative Feynman graphs [10] as well as to the analytic study of
nonperturbative classical solutions of string theory, including the nonperturbative vacuum of open
string theory [12].
An open problem is the construction of a regularized version of the BRST operator. The
regularization is indispensable to tame the associativity anomaly and to have a well defined theory.
Along with the successful regularization in MSFT, the BRST operator is also needed to insure
gauge invariance in the general formalism, and to be able to work outside of the Siegel gauge. In
particular, the BRST operator can be used to impose the additional gauge invariance conditions
in the Siegel gauge on the nonperturbative solutions we have obtained in [12]. Some of the issues
surrounding this problem are outlined following Eq.(4.8). These remarks apply not only to MSFT,
but also to any version of string theory that uses a cutoff of the string modes (including level
truncation), since the Virasoro algebra does not close with a finite number of modes. A substitute
for the Virasoro algebra at finite N, which tends to the Virasoro algebra at infinite N, is the key
to solving this problem.
Acknowledgments
I.K. would like to thank H. Hata, T. Kawano and K. Ohmori for valuable discussions and
comments. I.B. is supported in part by a DOE grant DE-FG03-84ER40168. I.K. is supported in
part by JSPS Research Fellowships for Young Scientists. Y.M. is supported in part by Grant-in-Aid
(# 13640267) from the Ministry of Education, Science, Sports and Culture of Japan.
A Brief review of MSFT in matter sector
A.1 Half-string for cosine modes
Here we review the split string formulation and its regularization which was developed in [8] and fix
notation in this paper. Although it was constructed to formulate the matter and the bosonized ghost
sector, we can apply the same formalism to fermionic functions which have a Fourier expansion in
terms of cosine mode in the full string basis.
A full string function φ(σ) which satisfies Neumann boundary conditions at the end points
d
dσφ(σ)
∣∣∣∣σ=0
=d
dσφ(σ)
∣∣∣∣σ=π
= 0 , (A.1)
43
has a Fourier expansion in terms of cosine modes
φ(σ) = φ0 +√2
∞∑
n=1
φn cosnσ , φ0 =1
π
∫ π
0dσφ(σ) , φn =
√2
π
∫ π
0dσφ(σ) cos nσ . (A.2)
Then split string functions l(σ), r(σ) for φ(σ) are defined as
φ(σ) =
l(σ) (0 ≤ σ ≤ π
2 )
r(π − σ) (π2 ≤ σ ≤ π), (A.3)
φ0 =1
π
∫ π2
0dσ(l(σ) + r(σ)) , φn =
√2
π
∫ π2
0dσ(l(σ) + (−1)nr(σ)) cos nσ . (A.4)
A.1.1 Dirichlet at midpoint
Split string functions with Neumann boundary conditions at the end points and Dirichlet boundary
conditions at the midpoint
l′(0) = r′(0) = 0 , l(π/2) = r(π/2) = φ (:= φ(π/2)) , (A.5)
have a Fourier expansion in terms of odd cosine modes o = 1, 3, 5, · · ·
l(σ) = φ+√2
∞∑
o=1
lo cos oσ , r(σ) = φ+√2
∞∑
o=1
ro cos oσ . (A.6)
The correspondence between φ, lo, ro and φe, φo is
φ = φ0 − wφe , lo = φo +Rφe , ro = −φo +Rφe , (A.7)
φ0 = φ+1
2v(lo + ro) , φe =
1
2T (lo + ro) , φo =
1
2(lo − ro) , (A.8)
where we used matrix notation for simplicity, and denoted
Roe =4
π
∫ π2
0dσ cos oσ
(cos eσ − cos
eπ
2
)=
4e2 io−e+1
πo(e2 − o2) ,
Teo =4
π
∫ π2
0dσ cos eσ cos oσ =
4o io−e+1
π(e2 − o2) , (A.9)
vo =2√2 io−1
πo=
1√2T0o , we =
√2 i−e+2 .
A.1.2 Neumann at midpoint
Split string functions with Neumann boundary conditions at the end points and Neumann boundary
conditions at the midpoint
l′(0) = r′(0) = 0 , l′(π/2) = r′(π/2) = 0 , (A.10)
44
are expanded in terms of even cosine modes e = 2, 4, 6, · · ·
l(σ) = φ+√2
∞∑
e=2
le (cos eσ − ie) , r(σ) = φ+√2
∞∑
e=2
re (cos eσ − ie) . (A.11)
The correspondence between split string and full string modes is
φ = φ0 − wφe , le = φe + Tφo , re = φe − Tφo , (A.12)
φ0 = φ+1
2w(le + re) , φe =
1
2(le + re) , φo =
1
2R(le − re) . (A.13)
A.1.3 Regularization
The infinite matrices T,R and vectors v,w defined in Eq.(A.10) satisfy the following relations
Roe = o−2Teoe2, Roe = Teo + vowe, vo =
∑
e>0
Teowe, we =∑
o>0
Roeve . (A.14)
As noted in [8], there is an ambiguity in naive computation using these matrices and vectors. For
example, T has an inverse matrix given by R, and yet it has a zero eigenvalue Tv = 0. This is
possible only because they are infinite dimensional matrices. It causes associativity anomalies. To
avoid the ambiguous results that come from the associativity anomaly, a finite matrix regularization
is proposed in [8]. We define N ×N matrices T,R and N -vectors v,w by
Roe = (κo)−2 Toe (κe)
2 , Roe = Toe + vowe, vo = Toewe, we = Reovo , (A.15)
where a bar means transpose, and we introduced a set of 2N frequencies κe, κo. These relations are
identical to the ones satisfied by the infinite matrices in Eq.(A.14), but we now use them as defining
relations for finite dimensional matrices and arbitrary frequencies. We can solve the equations in
Eq.(A.15) explicitly in term of the frequencies
Teo =wevoκ
2o
κ2e − κ2o, Roe =
wevoκ2e
κ2e − κ2o, (A.16)
we = i2−e
∏o′∣∣κ2e/κ2o′ − 1
∣∣ 12∏
e′ 6=e
∣∣κ2e/κ2e′ − 1∣∣ 12, vo = io−1
∏e′∣∣1− κ2o/κ2e′
∣∣ 12∏
o′ 6=o
∣∣1− κ2o/κ2o′∣∣ 12. (A.17)
By using only the defining relations we can show the following further relations for the regularized
version of T,R, v, w.
TR = 1e, RT = 1o, RR = 1 + ww, TT = 1− vv,T T = 1− ww
1 + ww, Tv =
w
1 + ww, vv =
ww
1 + ww, (A.18)
Rw = v(1 + ww), RR = 1 + vv (1 + ww) .
The original T,R, v, w in Eq.(A.10) are reproduced by setting the open string limit:
κe = e , κo = o , N →∞ . (A.19)
45
We note that at this limit ww diverges
1 + ww =
(N∏
n=1
κ2nκ2n−1
)2
→(√
πΓ (N + 1)
Γ(N + 1
2
))2
→ ∞ . (A.20)
A.2 Some results in matter sector
Here we summarize notation and conventions in the matter sector in MSFT.21
A.2.1 Oscillators in MSFT
• Mode expansion in matter sector (µ = 0, 1, · · · , d− 1):
Xµ(σ) = xµ0 +√2
∞∑
n=1
xµn cosnσ = xµ0 + i√2α′
∞∑
n=1
1
n(αµ
n − αµ−n) cos nσ , (A.21)
Pµ(σ) =1
π
(p0µ +
√2
∞∑
n=1
pnµ cosnσ
)=
1
π
(p0µ +
1√2α′
∞∑
n=1
ηµν(ανn + αν
−n) cosnσ
).
Nonzero modes in matter sector
αµn =√κna
µn , α
µ−n =
√κna
†µn , [αµ
n, ανm] = ǫ(n)κnδn+m,0η
µν , [aµn, a†νm ] = δn,mη
µν ,
xµn =i√2κn
ls(aµn − a†µn ) , pnµ =
√κn2
ηµνls
(aνn + a†νn ) , [xµn, pmν ] = iδn,mδµν ,
αn =1√2
(lsp|n| − iǫ(n)
κ|n|lsx|n|
), (A.22)
where we define the symbols l2s = 2α′, κn = n.
Zero mode in matter sector, with aµ0 := lspµ0
xµ0 =i
2ls√b(aµ0 − a†µ0 ) , p0µ =
1
ls√bηµν(a
ν0 + a†ν0 ) , [xµ0 , p0ν ] = iδµν , [a
µ0 , a
†ν0 ] = ηµν (A.23)
where b is some positive constant.
• Position eigenstates
〈x0, xn|xn = 〈x0, xn|xn , 〈x0, xn|x0 = 〈x0, xn|x0 ,xn|x0, xn〉 = xn|x0, xn〉 , x0|x0, xn〉 = x0|x0, xn〉, (A.24)
are given as squeezed states in Fock space
〈x0, xn| = 〈x0|e∑
n>0
(1
2κnα2n+
i√
2ls
xnαn− κn
2l2sx2n
)∏
n>0
(κnπl2s
) d4
21In this subsection, we use the same symbols for matter as we did for the ghosts in the main text for some
quantities, such as positions and momenta. We can avoid confusion from the context. We omit some definitions and
details because we can refer to Ref.[9] for them.
46
= 〈x0|e∑
n>0
(12a2n+
i√
2κnls
xnan− κn
2l2sx2n
)∏
n>0
(κnπl2s
) d4
,
〈x0| = 〈0|e12a20+i 2
ls√
bx0a0− 1
l2sbx20
(2
πl2sb
) d4
,
|x0, xn〉 =∏
n>0
(κnπl2s
) d4
e
∑n>0
(1
2κnα2−n− i
√2
lsxnα−n− κn
2l2sx2n
)
|x0〉
=∏
n>0
(κnπl2s
) d4
e
∑n>0
(12a†2n − i
√2κnls
xna†n− κn
2l2sx2n
)
|x0〉 ,
|x0〉 =
(2
πl2sb
) d4
e12a†20 −i 2
ls√
bx0a†0− 1
l2sbx20 |0〉 . (A.25)
They satisfy normalization and completeness conditions
〈x0, xn|x′0, x′n〉 = δd(x0 − x′0)∏
n>0
δd(xn − x′n) ,∫ddx0
∏
n>0
ddxn|x0, xn〉〈x0, xn| = 1 . (A.26)
• Oscillators as differential operators in position space
〈x0, xn|αm|Ψ〉 = −i√2
(ǫ(m)
κ|m|ls
x|m| + ls∂
∂x|m|
)〈x0, xn|Ψ〉 ,
〈x0, xn|α0|Ψ〉 = −ils∂
∂x0〈x0, xn|Ψ〉 . (A.27)
• Transformation from position space to Moyal space
A(x, xe, pe) = (det(2T ))d/2∫dxoe
− 2iθpeTxoΨ(x0, xn)
= 2Nd2 (1 + ww)−
d4
∫dxoe
− 2iθpeTxo〈x0, xn|Ψ〉 =: 〈x, xe, pe|Ψ〉 , (A.28)
where the midpoint xµ := Xµ (π/2) is related to the center of mass x0 through Eq.(A.21)
xµ := Xµ (π/2) = xµ0 −∑
e
xµewe, 〈x| = 〈x0| exp(ip ·∑
e
xewe
). (A.29)
The Moyal space 〈x, xe, pe| is given by a squeezed state
〈x, xe, pe| = 〈x|eα2e
2κe− α2
o2κo
−ξM0ξ−ξλdet(4κe1/2Tκ−1/2
o )d2
= 〈x|eα2e
2κe− α2
o2κo
−ξM0ξ−ξλ 2Nd(1 + ww)−d8 ,
M0 =
(κe
2l2s0
0 2l2sθ2Tκ−1
o T
), λ =
(−i
√2
lsαe − ipwe
−2√2lsθ Tκ−1
o αo
). (A.30)
47
• Moyal ⋆ product and trace:
⋆ = exp
(1
2
←−∂ξσ−→∂ξ
)= exp
(iθ
2
(←−∂xe
−→∂pe −
←−∂pe−→∂xe
)), σ = iθ
(0 1
−1 0
), (A.31)
TrA(x, ξ) = |det(2πσ)|− d2
∫dxedpeA(x, ξ) = (2πθ)−Nd
∫dxedpeA(x, ξ) . (A.32)
The normalization of a field A in Moyal space coincides with the normalization of its image
in Fock space (A.28)
〈Ψ|Ψ〉 =∫ddxTr
(A†(x, ξ) ⋆ A(x, ξ)
). (A.33)
• Oscillators as differential operators in Moyal space
〈x, xe, pe|α0|Ψ〉 = −ils∂
∂x〈x, xe, pe|Ψ〉 =: β0〈x, xe, pe|Ψ〉 ,
〈x, xe, pe|αe|Ψ〉 =(βxe −
w|e|√2β0
)〈x, xe, pe|Ψ〉 =
(βxe − w′
eβ0)〈x, xe, pe|Ψ〉
= βxe 〈x, xe, pe|Ψ〉 ,〈x, xe, pe|αo|Ψ〉 =: βpo〈x, xe, pe|Ψ〉 =
∑
e 6=0
βpe 〈x, xe, pe|Ψ〉U−e,o ,
βpo =∑
e>0
1√2
(ǫ(o)
θκ|o|2ls
R|o|e∂
∂pe+
2lsθpeTe|o|
)=∑
e 6=0
βpeU−e,o ,
βxe = − i√2
(ǫ(e)
κ|e|lsx|e| + ls
∂
∂x|e|
), βpe =
1√2
(θκ|e|2ls
ǫ(e)∂
∂p|e|+
2lsθp|e|
).(A.34)
These satisfy ordinary commutation relation:
[βxe , βxe′ ] = ǫ(e)κ|e|δe+e′ , [βpe , β
pe′ ] = ǫ(e)κ|e|δe+e′ , [βxe , β
pe′ ] = 0 ,
[βxe , βxe′ ] = ǫ(e)κ|e|δe+e′ , [βpo , β
po′ ] = ǫ(o)κ|o|δo+o′ , [βxe , β
po′ ] = 0 . (A.35)
• Oscillators as fields in Moyal space22
βxeA =
√κ|e|2
(βe ⋆ A−A ⋆ β−e) , βpeA =
√κ|e|2
(βe ⋆ A+A ⋆ β−e) ,
βe :=1√κ|e|
(− i
2lsǫ(e)κ|e|x|e| +
lsθp|e|
),
[βe, βe′ ]⋆ = ǫ(e)δe+e′ . (A.36)
We can also define odd mode fields through a Bogoliubov transformation
√κ|o|βo :=
∑
e 6=0
√κ|e|βeU−e,o . (A.37)
22The convention for βn here is the same as [12] βBKMn = βn, but differs by a factor from the convention in [9]
βBMn =
√κ|n|
2βn.
48
The following relations hold
βpoA =
√κ|o|2
(βo ⋆ A+A ⋆ β−o) ,
[βo, βo′ ]⋆ = ǫ(o)δo+o′ , [β−e, βo]⋆ = −ǫ(e)κ12
|e|U−e,oκ− 1
2
|o| . (A.38)
A.2.2 Butterfly projector
The momentum independent butterfly state AB(ξ) satisfies
βe ⋆ AB = AB ⋆ β−e = 0 , ∀e > 0 . (A.39)
There is a unique solution in monoid
AB(ξ) = 2dN exp
(−∑
e>0
(1
2l2sxeκexe +
2l2sθ2pe
1
κepe
)). (A.40)
It satisfies
AB ⋆ AB = AB , T r (AB) = 1. (A.41)
In ordinary oscillator language, Eq.(A.39) means
αe|ΨB〉 = 0 ,∑
o>0
(αoU
−1−o,e + α−oU
−1o,e
)|ΨB〉 = 0 , ∀e > 0 (A.42)
for zero momentum state. Now we take the ansatz
|ΨB〉 = N exp
−1
2
∑
m,n≥1
a†mVBmna
†n
|Ω〉 (A.43)
which corresponds to a monoid element in MSFT. Then we have constraints for V Bmn :
∑
o′>0
V Boo′√κo′U
−1−o′,e =
√κoU
−1o,e , o, e > 0 , V B
eo = V Boe = V B
ee′ = 0 . (A.44)
At the open string limit κe = e, κo = o,N = ∞, we can show that the matrix V Bmn which was
obtained in [27] :
V Bmn =
−(−1)m+n
2
√mn
m+n
Γ[m2]Γ[n
2]
πΓ[m+12
]Γ[n+12
]for m and n odd
0 for m or n even(A.45)
satisfies Eq.(A.44). Namely, we have obtained the correspondence:
AB ↔ |ΨB〉 = exp
(−1
2L−2
)|Ω〉 , for κe = e, κo = o,N =∞ . (A.46)
49
A.2.3 L0 and L0
In MSFT L0 in matter sector is defined as
L0 =1
2β20 +
∑
e>0
βx−eβxe +
∑
o>0
βp−oβpo
=1
2(1 + ww)β20 + β0
∑
e>0
ilswe∂
∂xe− d
2
∑
n>0
κn
+∑
e>0
(− l
2s
2
∂2
∂x2e− θ2
8l2sκ2e
∂2
∂p2e+
1
2l2sκ2ex
2e +
2l2sθ2p2e
)− 1
1 + ww
2l2sθ2
(∑
e>0
wepe
)2
. (A.47)
The operator L0 can be rewritten in terms of a field L0 using Moyal star product, plus a remnant
γ called the “midpoint correction term” [12], which is multiplied with an ordinary product
L0Aβ0 = L0(β0) ⋆ Aβ0 +Aβ0 ⋆ L0(−β0) + γAβ0 , (A.48)
L0(β0) :=∑
e>0
(l2sθ2p2e +
κ2e4l2s
x2e −lsθwepeβ0
)+
1
4(1 + ww)β20 −
d
4
∑
n>0
κn , (A.49)
γ = − 1
1 + ww
2l2sθ2
(∑
e>0
wepe
)2
. (A.50)
The γ term formally goes to zero as κn = n,N →∞, since ww →∞. However, this is not true in
computations due to contributions of the form ∞/∞ that are related to the associativity anomaly.
In fact, γ is indispensable to reproduce the correct spectrum of L0 [9][10]. The γ term depends
only on one special momentum mode p = (1 + ww)−1/2∑e>0wepe which we call the anomalous
midpoint momentum mode [9][11]. We can rewrite L0 in terms of oscillators
L0(β0) =∑
e>0
κeβ−e ⋆ βe +d
4
(∑
e>0
κe −∑
o>0
κo
)+
1
4(1 + ww)β20 −
lsθ(wepe)β0 , (A.51)
and then, acting on the butterfly projector (A.39), we have
L0AB = γAB . (A.52)
A.2.4 n-string vertex and Neumann coefficients
Here we give brief review of the correspondence of n-string vertex and Neumann coefficients in
MSFT. We note the properties of the momentum state and coherent states in Fock space, together
with the corresponding Moyal images.
• Momentum eigenstate (zero mode part):
|p0〉 =(2πbl2s
) d4 e−
12a†20 +
√blsa†0p0−
bl2s4p20 |0〉 ,
50
〈p0| = 〈0|e−12a20+
√blsa0p0− bl2s
4p20(2πbl2s
) d4 ,
〈p0|p′0〉 = (2π)dδd(p0 − p′0) ,〈p0|x0〉 = e−ip0x0 , 〈x0|p0〉 = eip0x0 . (A.53)
• Coherent state :23
〈Ψ|a† = 〈Ψ|µ∗ , 〈Ψ| = 〈p|eµ∗a ,2〈Ψ|V2〉12 = e−µ∗Ca†(1) | − p〉1 =: |Ψ〉1 ,A := 〈x, xe, pe|Ψ〉 = 2Nd(1 + ww)−
d8 e−ixpe
12µ∗2e − 1
2µ∗2o −ξM0ξ−ξλ ,
λ =
i√2
lsκ
12e µ∗e + ipwe
−2√2lsθ Tκ
− 12
o µ∗o
= 2K∗(µ∗ +Wp) ,
K∗ =
(ils
√κe
2 0
0 − lsθ T√
2κo
), W =
(ls√2κe
w
0
), (A.54)
where we used the reflector
〈V2| =
∫ddp(1)
(2π)dddp(2)
(2π)d〈0, p(1)|〈0, p(2)|e−
∑n≥1(−1)na
(1)n a
(2)n (2π)dδd(p(1) + p(2)) ,
|V2〉 =
∫ddp(1)
(2π)dddp(2)
(2π)d(2π)dδd(p(1) + p(2))e−
∑n≥1(−1)na
†(1)n a
†(2)n |0, p(1)〉|0, p(2)〉 . (A.55)
In particular we have the bra-ket correspondence for eigenstates of xn:
12〈V2|x0, xn〉2 = 1〈x0, (−1)nxn| . (A.56)
• Compute the n-string vertex for coherent states in terms of unknown Neumann coefficients
V rs(n), V
rs0(n), V
rs00(n)
|Vn〉 =
∫ddp(1)
(2π)d· · · d
dp(n)
(2π)d(2π)dδd(p(1) + · · ·+ p(n))
× e−12a†(r)V rs
(n)a†(s)−p(r)V rs
0(n)a†(s)− 1
2p(r)V rs
00(n)p(s) |p(i)〉 , (A.57)
〈Ψ1| · · · 〈Ψn|Vn〉 = (2π)dδd(p(1) + · · · + p(n))
× e−12µ(r)∗V rs
(n)µ(s)∗−p(r)V rs
0(n)µ(s)∗− 1
2p(r)V rs
00(n)p(s)
. (A.58)
• Compute the trace of the Moyal images of n coherent states in MSFT
∫ddxTr
(A1(x, ξ) ⋆ A2(x, ξ) ⋆ · · · ⋆ An(x, ξ)
)
= (−1)Nd2 (det((1 +m0)
n − (1−m0)n))−
d2 2nNd(1 + ww)−
nd8
× (2π)dδd(p(1) + · · ·+ p(n)) eE(n), (A.59)
23Here we introduce the bra coherent state. This is a different convention from that in [9].
51
E(n) = −1
2
∑
r,s
µ(r)∗C(2K∗−1m0O(s−r)(m0)K
∗ − δr,s)µ(s)∗
−2∑
r,s
p(r)WK∗−1m0O(s−r)(m0)K∗µ(s)∗ − 1
2
∑
r,s
p(r)(2WK∗−1m0O(s−r)(m0)K
∗W)p(s) ,
m0 :=M0σ =
(0 iθ
2l2sκe
−2il2sθ Tκ−1
o T 0
)(A.60)
where we used
CK∗−1m0 = −K∗σ , CK∗−1m0K∗ = −K∗−1m0K
∗C , O(s−r)(m0) = −O(r−s)(−m0) .
(A.61)
We note
m∗0 := K∗−1m0K
∗ =
0 −κ
12e Tκ
− 12
o
−κ−12
o T κ12e 0
= −m0 . (A.62)
The sign is changed compared to that in [9] because we used bra coherent state 〈Ψc| to define
the Moyal field A.
• The Neumann coefficients in the matter sector are obtained by identifying the Fock space
and MSFT expressions and comparing the exponents24
∫ddxTr
(A1(x, ξ) ⋆ A2(x, ξ) ⋆ · · · ⋆ An(x, ξ)
)= ρ〈Ψ1|〈Ψ2| · · · 〈Ψn|Vn〉 , (A.63)
ρ = (−1)Nd2 (det((1 +m0)
n − (1−m0)n))−
d2 2nNd(1 + ww)−
nd8 (A.64)
and using momentum conservation δd(p(1) + p(2) + · · · + p(n)). Then one has the Neumann
coefficients
V rs(n) = C
(2K∗−1m0O(s−r)(m0)K
∗ − δr,s),
V rs0(n) = −2K∗−1m0O(s−r)(−m0)K
∗W − 2
nW , (A.65)
V rs00(n) = 2WK∗−1m0O(s−r)(m0)K
∗W − 2
nWW .
They satisfy Neumann matrix algebra as in Ref.[9]. For the 3-string vertex we write them
explicitly
M(0) := CV rr(3) =
m∗20 − 1
m∗20 + 3
, M(±) := CV r,r±1(3) =
2± m∗0
m∗20 + 3
,
V(0) := V rr0(3) =
4m∗20
3(m∗20 + 3)
κ− 1
2e
wls√2, V(±) := V r,r±1
0(3) =−2m∗2
0 ∓ 6m∗0
3(m∗20 + 3)
κ− 1
2e
wls√2, (A.66)
V00 := V rr00(3) = l2s wκ
− 12
ett
tt+ 3κ− 1
2e w
24Here we defined the Witten’s ∗ product using the ket |V3〉 as Eq.(2.8) which is different convention from that in
[9]. Also, here we have included the overall normalization ρ which does not play a role in the computation of the
Neumann coefficients.
52
where we redefined as V r,s00 = V00δr,s using momentum conservation. We used the notation
w = (we, 0) , and t = κ1/2e Tκ
−1/2o .
• Fermionic ghost Neumann coefficients for the 3-vertex can be derived from matter Neumann
coefficients for the 6-vertex
Xrs := (−1)r+s√κn(V r,s(6) − V
r,s+3(6) )
1√κn
,
Xrs0 := (−1)r+s√κn(V r,s
0(6) − V r,s+30(6))l
−1s . (A.67)
This gives
X(0) = Cm∗2
0 − 1
3m∗20 + 1
, X(+) = −C 2m∗0 + 2m∗2
0
3m∗20 + 1
, X(−) = C2m∗
0 − 2m∗20
3m∗20 + 1
,
X(0)0 =
4m∗20
3m∗20 + 1
w√2, X
(+)0 =
2m∗0 − 2m∗2
0
3m∗20 + 1
w√2, X
(−)0 = −2m∗
0 + 2m∗20
3m∗20 + 1
w√2, (A.68)
where we defined
m∗0 :=
√κnm
∗0
1√κn
=√κnK
∗−1m0K∗ 1√
κn. (A.69)
B Derivation of regularized matrix formula
Here we sketch a derivation of fundamental formulas for regularized matrices presented in §2.1.3.We begin from the defining relations in Eq.(2.37). The first two equations imply Eq.(2.43) and
then from the remaining equations we have∑
o
Qeo(v′o)
2 = (κ′e)−1 ,
∑
e
Qeo(w′e)
2 = (κ′o)−1 , (B.1)
where
Qeo :=1
κ′e − κ′o, (B.2)
with e = ±2,±4, · · · ±2N, and o = ±1,±3, · · · ± (2N −1). Now we regard Q = (Qeo) as a 2N ×2N
matrix and compute its inverse
(Q−1)oe = (κ′e − κ′o)∏
o′ 6=o(κ′e − κ′o′)
∏e′ 6=e(κ
′o − κ′e′)∏
e′ 6=e(κ′e − κ′e′)
∏o′ 6=o(κ
′o − κ′o′)
. (B.3)
To prove the above formula, it is convenient to define a rational function f(z) which is determined
by the setup (N,κ′e, κ′o) uniquely:
f(z) :=
∏o′(z − κ′o′)∏e′(z − κ′e′)
. (B.4)
Next we compute (Q−1Q)oo′′ . We use contour integration and residues, where we denote the residue
of f(z) at z = z0 as Resz=z0f(z) and assume that the frequencies κ′e, κ′o are nondegenerate and
finite
∑
e
κ′e − κ′oκ′e − κ′o′′
∏o′ 6=o(κ
′e − κ′o′)
∏e′ 6=e(κ
′o − κ′e′)∏
e′ 6=e(κ′e − κ′e′)
∏o′ 6=o(κ
′o − κ′o′)
=∑
e
−Resz=κ′ef(z)
(κ′e − κ′o)(κ′e − κ′o′′)
∏e′(κ
′o − κ′e′)∏
o′ 6=o(κ′o − κ′o′)
53
= −∏
e′(κ′o − κ′e′)∏
o′ 6=o(κ′o − κ′o′)
∑
e
Resz=κ′e
f(z)
(z − κ′o)(z − κ′o′′)
=
∏e′(κ
′o − κ′e′)∏
o′ 6=o(κ′o − κ′o′)
∮
z=κ′o,κ′o′′
dz
2πi
f(z)
(z − κ′o)(z − κ′o′′)
=
∏e′(κ
′o − κ′e′)∏
o′ 6=o(κ′o − κ′o′)
Resz=κ′of(z)
(z − κ′o)2δo,o′′ = δo,o′′ . (B.5)
This shows that we have the correct inverse matrix Q−1.
Similarly, we can obtain (v′o)2, (w′
e)2 Eqs.(2.45,2.44) as follows:25
(v′o)2 =
∑
e
(Q−1)oe(κ′e)
−1 = −∏
e′(κ′o − κ′e′)∏
o′ 6=o(κ′o − κ′o′)
∑
e
1
κ′e
∏o′ 6=o(κ
′e − κ′o′)∏
e′ 6=e(κ′e − κ′e′)
= −∏
e′(κ′o − κ′e′)∏
o′ 6=o(κ′o − κ′o′)
∑
e
Resz=κ′ef(z)
z(z − κ′o)=
∏e′(κ
′o − κ′e′)∏
o′ 6=o(κ′o − κ′o′)
Resz=0f(z)
z(z − κ′o)
=
∏o′ 6=o κ
′o′∏
e′(κ′e′ − κ′o)∏
e′ κ′e′∏
o′ 6=o(κ′o′ − κ′o)
=
∏o′>0,o′ 6=|o| κ
2o′∏
e′>0(κ2e′ − κ2|o|)
2∏
e′>0 κ2e′∏
o′>0,o′ 6=|o|(κ2o′ − κ2|o|)
, (B.6)
(w′e)
2 =∑
o
(Q−1)oe(κ′o)
−1 =
∏o′(κ
′e − κ′o′)∏
e′ 6=e(κ′e − κ′e′)
∑
o
1
κ′o
∏e′ 6=e(κ
′o − κ′e′)∏
o′ 6=o(κ′o − κ′o′)
=
∏o′(κ
′e − κ′o′)∏
e′ 6=e(κ′e − κ′e′)
∑
o
Resz=κ′of(z)
z(z − κ′e)= −
∏o′(κ
′e − κ′o′)∏
e′ 6=e(κ′e − κ′e′)
Resz=0f(z)
z(z − κ′e)
=
∏e′ 6=e κ
′e′∏
o′(κ′e − κ′o′)∏
o′ κ′o′∏
e′ 6=e(κ′e − κ′e′)
=
∏e′>0,e′ 6=|e| κ
2e′∏
o′>0(κ2|e| − κ2o′)
2∏
o′>0 κ2o′∏
e′>0,e′ 6=|e|(κ2|e| − κ2e′)
(B.7)
where we assumed κ′e, κ′o are nonzero and Eqs.(2.42). We note that the above formula (B.6,B.7)
can also be rewritten as
(v′o)2 =
1
κ′oResz=κ′o
f(0)
f(z), (w′
e)2 =
1
κ′eResz=κ′e
f(z)
f(0). (B.8)
Now we consider the open string limit (A.19). By setting the open string limit κ′e = e, κ′o = o,N →∞, the rational function f(z) (B.4) becomes
f(z)
f(0)=
∏∞n=1(1− z2
(2n−1)2)
∏∞n=1(1− z2
4n2 )=(cos
πz
2
)( 2
πzsin
πz
2
)−1
=πz
2 tan πz2
. (B.9)
With this formula and Eq.(B.8) we can show that the regularized quantities reduce to the original
ones in Eq.(2.35).
C bc-ghost sector in position space
Here we consider the ghost position space representation of the n-point string vertices for n = 1, 2, 3,
starting with the Fock space formalism.25We choose the sign convention of w′e, v
′o such that they are consistent with Eq.(2.35).
54
Identity state
|I〉 =(∑
o>0
(−1) o−12 b−o
)(b0 + 2
∑
e>0
(−1) e2 b−e
)e∑∞
n=1(−1)n c−nb−n c0c1|Ω〉 , (C.1)
〈c0, xn, yn|I〉 =i√2
(∑
o>0
(−1) o−12 xo
)∏
e>0
((i√2xe)(
√2ye − 2(−1) e
2 c0)). (C.2)
This is BRST invariant[2] although the form is rather complicated.
Reflector
12〈V2| = 1〈Ω|c(1)−1 2〈Ω|c(2)−1e−∑∞n=1(−1)n(c
(1)n b
(2)n +c
(2)n b
(1)n )(c
(1)0 + c
(2)0 ) , (C.3)
|V2〉12 = (b(1)0 − b
(2)0 )e
∑∞n=1(−1)n(c
(1)−n b
(2)−n+c
(2)−nb
(1)−n)c
(1)0 c
(1)1 |Ω〉1 c
(2)0 c
(2)1 |Ω〉2 , (C.4)
1〈c(1)0 , x(1)n , y(1)n |2〈c(2)0 , x(2)n , y(2)n |V2〉12 = (c(1)0 + c
(2)0 )
∞∏
n=1
(−2i((−1)nx(1)n + x(2)n )((−1)ny(1)n + y(2)n )
)
= 〈c(1)0 , x(1)n , y(1)n | − c(2)0 ,−(−1)nx(2)n ,−(−1)ny(2)n 〉 . (C.5)
3-string vertex
|V3〉123 = e∑3
r,s=1
(−c†(r)Xrsb†(s)−c†(r)Xrs
0b(s)0
)
c(1)0 c
(1)1 |Ω〉1c
(2)0 c
(2)1 |Ω〉2c
(3)0 c
(3)1 |Ω〉3 , (C.6)
1〈c(1)0 , x(1)n , y(1)n |2〈c(2)0 , x(2)n , y(2)n |3〈c
(3)0 , x(3)n , y(3)n |V3〉123
= − detr,s,n,m
(δr,sδn,m +Xrsnm)(c
(1)0 − wy(1)e )(c
(2)0 − wy(2)e )(c
(3)0 − wy(3)e )ei
∑r,s y
(r)( 1−X1+X )
rsx(s)
(C.7)
where we used the relation[21][22][23],
Xrs0 = (δrs +Xrs)
w√2. (C.8)
Witten’s star product in ghost position space
Ψ1 ⋆W Ψ2(c0, xn, yn) =
∫dc
(2)0 dc
(3)0
dx(2)n dy
(2)n
−2idx
(3)n dy
(3)n
−2i×1〈c0, xn, yn|2〈
˜c(2)0 , x
(2)n , y
(2)n |3〈
˜c(3)0 , x
(3)n , y
(3)n |V3〉123Ψ1(c
(2)0 , x(2)n , y(2)n )Ψ1(c
(3)0 , x(3)n , y(3)n )
=
∫dc
(2)0 dc
(3)0
dx(2)n dy
(2)n
−2idx
(3)n dy
(3)n
−2i 1〈c0, xn, yn|2〈c(2)0 , x(2)n , y(2)n |3〈c(3)0 , x(3)n , y(3)n |V3〉123
×Ψ1(−c(2)0 ,−(−1)nx(2)n ,−(−1)ny(2)n )Ψ2(−c(3)0 ,−(−1)nx(3)n ,−(−1)ny(3)n ) (C.9)
where
1〈 ˜c0, xn, yn| := 12〈V2|c0, xn, yn〉2 = 1〈−c0,−(−1)nxn,−(−1)nyn| . (C.10)
55
D Moyal ⋆ for (DD)b (NN)c split strings
In this appendix, we examine the remaining choice of the midpoint boundary condition for the
split string variables as compared with our discussion in section 2.3. Namely, we consider Dirichlet
boundary condition for b(σ) and Neumann for c(σ) at the midpoint σ = π/2. In this case the left
and right half of b(σ) : lb(σ), rb(σ) satisfy Dirichlet boundary condition at both σ = 0, π/2, and
those of c(σ) : lc(σ), rb(σ) satisfy Neumann at σ = 0, π/2. The lb(σ), rb(σ) and lc(σ), rc(σ) are
expanded in terms of even sine/cosine modes respectively
lb(σ) =2
πσb+ i
√2
∞∑
e=2
lbe sin eσ , rb(σ) =2
πσb+ i
√2
∞∑
e=2
rbe sin eσ , (D.1)
lc(σ) = c+√2
∞∑
e=2
lce(cos eσ − ie) , rc(σ) = c+√2
∞∑
e=2
rce(cos eσ − ie) . (D.2)
From Eqs.(2.69)(2.70)(2.27) (A.12), we have relations between split and full string variables:
b = ¯wxo , lbe = xe + T xo , rbe = −xe + T xo , (D.3)
c = c0 − wye , lce = ye + Tyo . rce = ye − Tyo . (D.4)
With this setup Witten type product in the split string formulation becomes:
A′ ∗ B′(b, c, lbe, lce, r
be, r
ce) =
∫ ∏
e>0
(idηbedη
ce
)A′(b, c, lbe, l
ce, η
be, η
ce)B
′(b, c, ηbe,−ηce, rbe, rce) . (D.5)
where the split string and full string fields in position space are the same
A′(b, c, lbe, lce, r
be, r
ce) ∼ Ψ(c0, xn, yn) (D.6)
by substituting on the right hand side the inverse maps obtained in Eqs.(D.3)(D.4)
xe =1
2(lbe − rbe) , xo = uob+
1
2Soe(l
be + rbe) , (D.7)
c0 = c+1
2we(l
ce + rce) , ye =
1
2(lce + rce) , yo =
1
2Roe(l
ce − rce) . (D.8)
These relations are valid only when ww =∞ in the limit: κe = e, κo = o,N =∞.
As the next step, we consider the Moyal formulation, including the regularization with (N,κe, κo).
Comparing Eqs.(D.3)(D.4)(D.5) with Eq.(2.63), we identify
x ∼ T xo , y ∼ 2xe , x′ ∼ −Tyo , y′ ∼ 2ye ,
and define new variables with even index E by
xE := T xo , yE := Tyo . (D.9)
56
At the limit κe = e, κo = o,N = ∞, T has a zero mode u (2.34) and we meet as usual the
associativity anomaly, but at finite N everything is well-defined. From Eq.(2.63)(D.6) we obtain
the Moyal image A′(b, c, xE, pE , yE , qE) of the position space field Ψ(c0, xn, yn) :
A′(b, c, xE , pE , yE, qE)
= 2−2N
∫ ∏
e>0
(i−1dxedye
)e−2pExe−2qEyeA′(b, c, xe + xE , ye + yE,−xe + xE , ye − yE)
= 2−2N
∫ ∏
e>0
(i−1dxedye
)e−2pExe−2qEyeΨ(c+ wye, xe, ub+ SxE, ye, RyE) , (D.10)
and the corresponding Moyal ⋆ product becomes
⋆ = e− 1
2
( ←−∂
∂xE
−→∂
∂pE+←−∂
∂yE
−→∂
∂qE+←−∂
∂pE
−→∂
∂xE+←−∂
∂qE
−→∂
∂yE
)
. (D.11)
In this case, the above formula is more complicated than our previous choice due to the additional
b-ghost midpoint mode b.
E Ghost butterfly projector with even modes
The butterfly projector in Eq.(2.126) is based on the odd mode oscillators in Eq.(2.10). There is
another choice using even mode oscillators Eq.(2.140)
βbe ⋆ A′B = βce ⋆ A
′B = A′
B ⋆ βb−e = A′
B ⋆ βc−e = 0 , ∀e > 0 . (E.1)
Explicitly, we have the even butterfly state
A′B = ξ0 2
−2N exp
(−∑
e>0
(ixbeκex
ce +
4i
θ′2pbeκ
−1e pce
))(E.2)
in the Siegel gauge.
As we will see in the following, the even butterfly is the one defined by Gaiotto-Rastelli-Sen-
Zwiebach(GRSZ) using twisted ghosts [25]. The conditions in Eq.(E.1) correspond to
be|ΨB〉 = 0 , ce|ΨB〉 = 0 ,∑
o>0
(boU
−1−o,e + b−oU
−1o,e
)|ΨB〉 = 0 ,
∑
o>0
(Ue,−oco + Ue,oc−o) |ΨB〉 = 0 (E.3)
for all e > 0 in ordinary oscillator language. If we take a gaussian ansatz
|ΨB〉 = N exp
∑
n,m≥1
c−mVBmnb−n
c1|Ω〉 , (E.4)
the above conditions become
V Beo = V B
oe = V Bee′ = 0 , e, e′, o > 0 , (E.5)
57
∑
o′>0
V Bo′oU
−1−o′,e = −U−1
o,e , e > 0 , (E.6)
∑
o′>0
V Boo′Ue,−o′ = Ue,o , e > 0 . (E.7)
We can solve Eq.(E.6) by comparing it with the matter one (A.44) as
V Bmn = − 1√
nV Bnm
√m = −√mV B
mn
1√n
=
(−1)m+n2
mm+n
Γ[m2]Γ[n
2]
πΓ[m+12
]Γ[n+12
]for m and n odd
0 for m or n even. (E.8)
We can check that this V B satisfies Eq.(E.7). The ghost butterfly which we have obtained in MSFT
as above can be identified with GRSZ’s (twisted) ones [25]26:
A′B ↔ |ΨB〉 = exp
(−1
2L′−2
)|Ω′〉 for κe = e, κo = o,N =∞ .
The relation between the matter and (twisted) ghost generating functions is obtained by using [28]
∂
∂zS(w, z) = S(z, w)
where we defined the generating functions
S(z, w) :=∞∑
m,n=1
√mn(−z)m−1(−w)n−1Smn =
1
(z −w)2 −f ′(z)f ′(w)
(f(z)− f(w))2 , (E.9)
S(z, w) :=
∞∑
m,n=1
(−z)m−1(−w)nSmn = − w
z(w − z) +f(w)
f(z)
f ′(z)f(w)− f(z) , (E.10)
|S〉 = exp
−1
2
∞∑
m,n=1
a†mSmna†n
|Ω〉 , |S〉 = exp
∞∑
m,n=1
c−mSmnb−n
|Ω′〉 (E.11)
namely
Smn = − 1√nSnm√m = −√mSmn
1√n. (E.12)
Here we have V Bmn = Smn for the conformal mapping f(z) = z√
1+z2which represents (canonical)
butterfly state e−12L−2 |Ω〉 [27] and the relation Eq.(E.8).
F Algebra of gaussian operators
We discuss algebraic relations for gaussians constructed from bosonic and fermionic oscillators
which we used to compute the propagator (4.17) in MSFT.
26There is a correspondence for the vacuum: c1|Ω〉 ∼ |Ω′〉 .
58
For bosonic oscillators: a, a†, [a, a†] = 1, we can prove a formula
ea†Aa†+aBa = e−
12Tr log(cos(2
√AB))e
12a† tan(2
√AB)
√ABB−1a†
× e−a† log(cos(2√AB))ae
12aB
tan(2√
AB)√AB
a(F.1)
using a similar method to Appendix A in [32]. Here A,B are symmetric matrices: A = A, B = B.
Then we have
eηAη+ ∂∂η
B ∂∂η eiξη = e−
12Tr log(cos(2
√AB))
× e12η tan(2
√AB)
√ABB−1η− 1
2ξB
tan(2√
AB)√AB
ξ+iη 1
cos(2√
AB)ξ
(F.2)
where we used the relation[∂
∂η, η
]= 1, e
ηC ∂∂η eiξηe
−ηC ∂∂η = eηe
Cξ. (F.3)
We obtain the formula for the propagator∫dMξ e−iξηe
η′Aη′+ ∂∂η′B
∂∂η′ eiξη
′
= (2π)M2 e
− 12Tr log
(B
sin(2√
AB)√AB
)
e− 1
2η
√AB
tan(2√
AB)B−1η− 1
2η′
√AB
tan(2√
AB)B−1η′+η′
√AB
sin(2√
AB)B−1η
. (F.4)
When the momentum is nonzero in (4.17), we need the modified version of the above formula:∫dMξ e−iξηe
η′Aη′+ ∂∂η′B
∂∂η′+Cη′
eiξη′
= (2π)M2 e
− 12Tr log
(B
sin(2√
AB)√AB
)
e− 1
2η
√AB
tan(2√
AB)B−1η− 1
2η′
√AB
tan(2√
AB)B−1η′+η′
√AB
sin(2√
AB)B−1η
× e−14CA−1
(1− tan
√AB√
AB
)C+ 1
2(η+η′) tan
√AB√
ABC. (F.5)
For fermionic oscillators: a, a†, ai, a†j = δij , we have a similar formula
ea†Aa†+aBa = e
12Tr(log cosh(2
√AB))e
12a† tanh(2
√AB)
√ABB−1a†
× e−a† log(cosh(2√AB))ae
12aB
tanh(2√
AB)√AB
a, (F.6)
where A,B are antisymmetric matrices A = −A, B = −B. Notingη,
∂
∂η
= 1 , e
ηC ∂∂η e−ξηe
−ηC ∂∂η = eηe
Cξ , (F.7)
we obtain
eηAη+ ∂
∂ηB ∂
∂η e−ξη = e12Tr(log cosh(2
√AB))e
12η tanh(2
√AB)
√ABB−1η+ 1
2ξB tanh 2
√AB√
ABξ+η 1
cosh 2√
ABξ.
By integration we have∫dξ eξηe
η′Aη′+ ∂∂η′B
∂∂η′ e−ξη′
= det12
(Bsinh 2
√AB√
AB
)e
12η
√AB
tanh 2√
ABB−1η+ 1
2η′
√AB
tanh 2√
ABB−1η′−η′
√AB
sinh 2√
ABB−1η
. (F.8)
59
G Neumann coefficients
G.1 Neumann coefficients from CFT
In this subsection, we give a short summary of the analytic expression of Neumann coefficients
given in [2] which are obtained from conformal field theory. We introduce a set of numbers An, Bn
(n = 0, 1, 2, · · · ) which appear in the Taylor expansion,(1 + ix
1− ix
)1/3
=∑
e≥0
Aexe + i
∑
o>0
Aoxo ,
(1 + ix
1− ix
)2/3
=∑
e≥0
Bexe + i
∑
o>0
Boxo . (G.1)
From these data, the Neumann coefficients (N(0,±)nm for matter sector and N
(0,±)nm for ghost sector)
are written as follows. First when n,m > 0 and n 6= m,
N (0)nm =
(−1)n
3
(AnBm+BnAm
(n+m) + AnBm−BnAm
(n−m)
)n+m = even
0 n+m = odd, (G.2)
N (±)nm =
−(−1)n
6
(AnBm+BnAm
(n+m) + AnBm−BnAm
(n−m)
)n+m = even
±√3
6
(AnBm−BnAm
(n+m) + AnBm+BnAm
(n−m)
)n+m = odd , (G.3)
N (0)nm =
−(−1)n
3
(AnBm+BnAm
(n+m) − AnBm−BnAm
(n−m)
)n+m = even
0 n+m = odd, (G.4)
N (±)nm =
(−1)n
6
(AnBm+BnAm
(n+m) − AnBm−BnAm
(n−m)
)n+m = even
∓√3
6
(AnBm−BnAm
(n+m) − AnBm+BnAm
(n−m)
)n+m = odd . (G.5)
For the diagonal components (n = m > 0), they are replaced by,
N (0)nn =
1
3n
(2(−1)n(1 +
n∑
k=1
(−1)kA2k)− (−1)n −A2
n
), N (±)
nn = −(−1)n2n
− N(0)nn
2, (G.6)
N (0)nn = N (0)
nn −2(−1)nAnBn
3n, N (±)
nn = −(−1)n2n
− 1
2N (0)
nn . (G.7)
For the zero mode, we use
N(0)0m =
23mAm m = even
0 m = odd, N
(±)0m =
−13mAm m = even∓√3
3m Am m = odd, N00 = −
1
2ln33
42, (G.8)
N(0)0m =
−23mBm m = even
0 m = odd, N
(±)0m =
13mBm m = even∓√3
3m Bm m = odd. (G.9)
There are some differences in the convention to make direct comparison of these quantities with
the corresponding ones obtained in Moyal language which are given in (A.66, A.68). We summarize
them as follows,
M(0,±)nm (cft) ≡ −(−1)n√mnN (0,±)
mn , V(0,±)n (cft) ≡ −ls
√nN
(0,±)0n , V00(cft) ≡ −l2sN00 ,(G.10)
60
X(0,±)nm (cft) ≡ mN (0,±)
nm , X(0,±)m0 (cft) ≡ mN (0,±)
0m . (G.11)
The sign factor inM(0,±)(cft) comes in because we include the multiplication of Cnm = (−1)nδn,min the MSFT definition.
G.2 Ratios of MSFT-regulated and CFT Neumann coefficients
The MSFT-regulated Neumann coefficients are discussed in sections 3.2,3.3 and A.2.4. The numer-
ical ratiosM(0)ee (N) /M(0)
ee′ (cft) and X(0)ee (N) /X
(0)ee′ (cft) for e, e
′ = 2, 4, 6, 8 at N = 5, 20, 100, 400,
shows the convergence of these to the CFT values in the large N limit.
M(0)
ee′ (5)
M(0)
ee′ (cft)2 4 6 8
M(0)
ee′ (20)
M(0)
ee′ (cft)2 4 6 8
2 1.15355 1.27359 1.43938 1.71214 2 1.02373 1.04035 1.05899 1.07957
4 1.27359 1.41879 1.61307 1.92691 4 1.04035 1.05982 1.08099 1.10391
6 1.43938 1.61307 1.84084 2.20463 6 1.05899 1.08099 1.10438 1.12937
8 1.71214 1.92691 2.20463 2.64473 8 1.07957 1.10391 1.12937 1.15628
M(0)
ee′ (100)
M(0)
ee′ (cft)2 4 6 8
M(0)
ee′ (400)
M(0)
ee′ (cft)2 4 6 8
2 1.00272 1.00459 1.00664 1.00885 2 1.00043 1.00071 1.00103 1.00137
4 1.00459 1.00675 1.00907 1.01151 4 1.00071 1.00105 1.0014 1.00178
6 1.00664 1.00907 1.0116 1.01426 6 1.00103 1.0014 1.00179 1.0022
8 1.00885 1.01151 1.01426 1.01709 8 1.00137 1.00178 1.0022 1.00263
X(0)
ee′ (5)
X(0)
ee′ (cft)2 4 6 8
X(0)
ee′ (20)
X(0)
ee′ (cft)2 4 6 8
2 1.28946 1.42714 1.60523 1.89334 2 1.1113 1.15228 1.19097 1.22861
4 1.42714 1.56675 1.75409 2.06284 4 1.15228 1.18898 1.22491 1.26065
6 1.60523 1.75409 1.9584 2.29906 6 1.19097 1.22491 1.25898 1.29343
8 1.89334 2.06284 2.29906 2.69597 8 1.22861 1.26065 1.29343 1.32697
X(0)
ee′ (100)
X(0)
ee′ (cft)2 4 6 8
X(0)
ee′ (400)
X(0)
ee′ (cft)2 4 6 8
2 1.03837 1.052 1.06441 1.07597 2 1.01529 1.02071 1.02562 1.03019
4 1.052 1.06393 1.07517 1.08587 4 1.02071 1.02544 1.02988 1.03409
6 1.06441 1.07517 1.08556 1.09557 6 1.02562 1.02988 1.03397 1.0379
8 1.07597 1.08587 1.09557 1.10504 8 1.03019 1.03409 1.0379 1.0416
61
H Twist and SU(1, 1) in the Siegel gauge
The twist operator is usually given by Ω = (−1)L0 . In MSFT this becomes
βΩA(x, xe, pe; ξ0, xo, po, yo, qo) = A(x, xe,−pe; ξ0,−xo, po,−yo, qo) . (H.1)
This follows from
⟨x0, xe, xo; c0, x
ghe , x
gho , y
ghe , ygho
∣∣∣ Ω =⟨x0, xe,−xo; c0, xghe ,−xgho , yghe ,−ygho
∣∣∣ . (H.2)
When we use the even variables xbe, pbe, x
ce, p
ce, we have the expression
βΩA(x, xe, pe; ξ0, xbe, p
be, x
ce, p
ce) = A(x, xe,−pe; ξ0,−xbe, pbe,−xce, pce) . (H.3)
The SU(1, 1) generators are given by
G :=∞∑
n=1
(c−nbn − b−ncn) (= Ngh − (c0b0 + 1)) ,
X := −∞∑
n=1
nc−ncn , Y :=
∞∑
n=1
1
nb−nbn (H.4)
by oscillator representation [30][31]. In MSFT we translate them as
βG =∑
o>0
(yo
∂
∂yo− xo
∂
∂xo+ po
∂
∂po− qo
∂
∂qo
)=∑
e>0
(xce
∂
∂xce− xbe
∂
∂xbe+ pbe
∂
∂pbe− pce
∂
∂pbe
),
βX = i∑
o>0
(yoκo
∂
∂xo− poκo
∂
∂qo
)= i∑
e>0
(xce
∂
∂xbe− pbe
∂
∂pce
), (H.5)
βY = i∑
o>0
(xoκ
−1o
∂
∂yo− qoκ−1
o
∂
∂po
)= i∑
e>0
(xbe
∂
∂xce− pce
∂
∂pbe
)
on the fields in the Siegel gauge. These operators satisfy the su(1, 1) algebra:
[βX , βY ] = −βG , [βG , βX ] = 2βX , [βG , βY ] = −2βY . (H.6)
They are derivations with respect to the Moyal ⋆ product:
βO(A1 ⋆ A2) = (βOA1) ⋆ A2 +A1 ⋆ (βOA2) , O = G, X, Y . (H.7)
In fact, the above su(1, 1) generators (H.5) are inner derivations
βOA = [βO, A]⋆ , O = G, X, Y ,
βG =1
θ′∑
o>0
(yoqo − xopo) =1
θ′∑
e>0
(xcep
ce − xbepbe
), (H.8)
βX =i
θ′∑
o>0
yoκopo =i
θ′∑
e>0
xcepbe , βY =
i
θ′∑
o>0
xoκ−1o qo =
i
θ′∑
e>0
xbepce .
62
By Eqs.(H.3,H.7), we can restrict solutions of the equations of motion (4.15) to the twist even and
SU(1, 1) singlet sector:
βΩA(ξ) = A(ξ) , βOA(ξ) = 0 , O = G, X, Y (H.9)
in the Siegel gauge consistently. In fact, we note that in the Siegel gauge
[βO, L0] = 0 , O = Ω, G, X, Y ,βΩ(A ⋆ A) = (βΩA)⋆(βΩA) = (βΩA) ⋆ (βΩA) , ⋆ := ⋆|−θ,−θ′ (H.10)
from Eqs.(4.10,4.11). This condition (H.9) can be used to search for the nonperturbative tachyon
vacuum.[30]
In MSFT, it is convenient to note the monoid structure (§3). It consists of gaussian Moyal
fields: AN ,M,λ = N e−ξMξ−ξλ. We can consider the twist and SU(1, 1) symmetric class within the
monoid. From Eqs.(H.3,H.5) the restriction by this symmetry (H.9) of a monoid element AN ,M,λ(ξ)
is given by
M = εM ′ :=
(0 M ′
−M ′ 0
), M ′ =
(A 0
0 B
), A = A , B = B , λ = 0 (H.11)
in the basis ξ = (xbe, pbe, x
ce, p
ce)
27. Namely, the coefficient matrix of the quadratic term in the
exponent becomes block diagonal and symmetric. For example, the perturbative vacuum and
butterfly states (4.22,E.2) are of the form of (H.11). Their τ -evolved gaussians are also in this
class (4.31). We note that this class of gaussian is not closed within the monoid because of twist
operator which changes the sign of noncommutative parameters θ, θ′ in the Moyal ⋆ product, while
the SU(1, 1)-symmetry is conserved in the Siegel gauge by Eq.(H.7).
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